As I see it, there are two basic parts to this question: can you hold the charge on a Reach spell, and is the 'ray' made by a Reach spell visible?
To the first question: You cannot hold the charge on a Reach spell. A Reach spell is a ranged touch attack, and the rules for holding the charge specifically call out touch attacks, which are different. The relevant rules are:
Touch
You must touch a creature or object to affect it. A touch spell
that deals damage can score a critical hit just as a weapon can. A
touch spell threatens a critical hit on a natural roll of 20 and deals
double damage on a successful critical hit. Some touch spells allow
you to touch multiple targets. You can touch as many willing targets
as you can reach as part of the casting, but all targets of the spell
must be touched in the same round that you finish casting the spell.
Touch Spells and Holding the Charge:
In most cases, if you don’t discharge a touch spell on the round you cast it, you can hold the charge (postpone the discharge of the spell) indefinitely. You can make touch attacks round after round. If you cast another spell, the touch spell dissipates.
Since that passage only mentions touch spells, and not ranged touch spells, you cannot hold the charge on a Reach spell.
To the second question: Whether or not you can stealthily cast a Reach spell is pretty much entirely up to your GM. The description of Reach Spell says that the affected spell "effectively becomes a ray", which says to me that for all intents and purposes, you treat the spell like a ray.
The fact that it says "effectively becomes a ray" and not "becomes a ray" is a semantic difference that has no real meaning. There are no exclusions listed for why it says one and not the other, so it's likely that whatever designer wrote this feat liked the sound of "effectively" more than not, and his/her editor agreed. Since the feat doesn't list any ways that the spell doesn't work like a ray, it's a reasonable assumption that it always works like a ray.
If your GM says that Reach Spell rays are visible, then they are. The rules aren't terribly clear on whether rays in general are visible, but clues from flavour text and descriptions strongly imply that ray spells are visible to normal vision, and thus can be seen by anyone who is looking.
There is no further clarification in the text, but most of what you're looking for is in the quotes above.
The Target of a spell, as well as it's Area of Effect are both rules-defined terms.
Target or Targets: Some spells have a target or targets. You cast these spells on creatures or objects, as defined by the spell itself. You must be able to see or touch the target, and you must specifically choose that target. You do not have to select your target until you finish casting the spell.
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Area: Some spells affect an area. Sometimes a spell description specifies a specially defined area, but usually an area falls into one of the categories defined below.
Regardless of the shape of the area, you select the point where the spell originates, but otherwise you don't control which creatures or objects the spell affects. The point of origin of a spell is always a grid intersection. When determining whether a given creature is within the area of a spell, count out the distance from the point of origin in squares just as you do when moving a character or when determining the range for a ranged attack. The only difference is that instead of counting from the center of one square to the center of the next, you count from intersection to intersection.
So if you are choosing to target a creature that you can see or touch with a spell that can target creatures, that counts as an 'attack' for the purposes of invisibility. Even if the spell is (Harmless) or otherwise not an 'attack'. If you similarly target a spell with an area of effect on a square intersection that includes a creature inside it's area of effect (as defined above), that counts as an 'attack' for the purposes of invisibility. Even if the spell is (Harmless) or otherwise not an attack. Alarm, for example, breaks invisibility.
Further, the rules of invisibility do not specify casting a spell that fulfills those terms. If you cast a spell, and it is still in effect three rounds later, and an enemy enters it's existing area of effect? Your invisibility breaks. For example, if you cast obscuring mist on empty space, and an enemy enters it a few rounds later.
So grease, a spell that has a duration, would cause your invisibility to break if enemies entered it's area of effect before it's expiration. If it was an instantaneous creation spell, enemies entering the area of the spell after it's cast would not break your invisibility, as the spell would have ended and the grease remaining would be mundane. However, if you coated an object in grease, you are neither creating an area of effect nor targeting any creature, so you do not break your invisibility, even though doing so causes a foe to make a reflex save - you are not targeting any foes, nor does any area of effect of your spell(s) include a foe.
Web similarly has an area of effect and a duration. Until that duration is finished, the area of effect remains and if that area of effect 'includes a foe', which it would at the end of the action that caused the movement that placed the foe inside the aoe, your invisibility breaks. Expeditious excavation would not break your invisibility, even if you tunneled a pit and the enemy fell into it - by the time they fell into it, the spell would have ended, and the spell never included them in an aoe or targeted them. The exception to this is if you targeted a square of dirt with a earth gliding foe inside it, or targeted a square with them in it in some other manner.
However
Note that strict definitions of this type are often ignored in actual play, as GMs tend to use logic to define 'direct' and 'indirect' harm that doesn't directly reference the specific rules terms.
RAI
There may be forum posts or other trivia from designers on this topic, but I do not follow the paizo forums or developers. Further, such PF dev posts often contradict rules text or display ignorance of the existing rules, making 'RAI' very hit and miss and unreliable, especially for cases like this.
In the End
Invisibility's 'ends if the character causes harm to a foe' clause has historically been the cause of a lot of contention, which is assumedly why the designers of pathfinder modified the original description in the attempt to make it clear-cut. However, ultimately, the degree of 'screwing with people while invisible' your table wishes to allow will vary based on the attitudes of the players and the GM.
Some tables believe it is a staple of the trope, and interpret the rules very liberally as regards screwing with people while invisible. Others bend rules (to the point of largely ignoring the RAW) and 'break' invisibility during actions that are, by RAW, entirely fine.
As this varies widely by table, the way in which this rule will be interpreted varies just as widely. While you could write a definitive set of rules for allowing indirect action while penalizing direct action, it would be cumbersome, and less useful than simply having a GM on the same page as you about what an invisible character can and can't do without breaking the spell (or alerting foes as to their location).
Best Answer
No, you have to steal the orb, which ends invisibility
They are in combat, to search something you have to spend at least a Move Action to locate it before using Take Something Unnoticed assuming it's visible to you, because if not, you have to search the target looking for it (similar to looking for a hidden/secret door), which requires at least a Touch attack if your target is mid-combat.
Trying to take something during combat would require a Steal Maneuver, assuming the item is visible and not hidden too, with a bonus from being invisible (+2) and a penalty on the target's CMD as they are unaware of you (losing Dex bonus to AC/CMD), which would then break the spell, being an attack against CMD.
Enemies are not stationary in combat, they are moving, dodging, jumping and swinging their weapons, and you are attempting to search through their pockets for an orb, while they are doing all that and you can't even tell the location of this orb by sight.
Even the Steal maneuver isn't enough to take something that is inside a bag or or backpack. There is a feat for that: Graceful Steal.
Similarly, the Swipe and Stash feat says that you must use the Steal maneuver when trying to use the feat to plant an item on an enemy in combat, instead of a simple Sleight of Hand check against Perception. Though this text is exclusive to the feat, the same logic can be applied to take something unnoticed.
What you have to keep in mind that, even though you are not attacking or defending yourself, you are in combat as well, you have to roll initiative as soon as your targets and the orcs do, and you gotta act on your initiative order, regardless if they are aware of your presence or not.
Weren't your targets in combat, as the GM, I would allow you to make multiple checks looking for it, depending on how many pockets those enemies got, and depending on whether or not your information about said orb being really inside their robe pockets is correct, as it could be hidden inside a belt pocket, or even in their backpack inside a small chest, etc.
On each of those checks, you first would have to succeed at a Stealth check, to search for things unnoticed, and a Perception check, to use your hands to find it. On each Stealth check, the enemies would be allowed an opposed Perception check to notice your hands shuffling through their things and body. Your own Perception check wouldn't have a high DC, as you know exactly what you are looking for (again, unless your information was false).
This wouldn't break invisibility, as you are not causing harm to your targets in any way.
To be honest, you are probably better first using Detect Magic (since its a magic orb) to find this orb, or waiting for each enemy to be taken down during combat and then search his body, as their enemies (the orcs) won't simply stop mid-combat to search them for the orbs while there is a risk of getting attacked by her enemies.