Shadow Cloak

Activation: — and immediate (command)

The information above is pulled directly from the items info. We see that it takes an immediate action to activate either of its command effects.

For reference,

Swift Action A swift action consumes a very small amount of time, but represents a larger expenditure of effort and energy than a free action. You can perform only a single swift action per turn.

Immediate Action An immediate action is very similar to a swift action, but can be performed at any time — even if it's not your turn.

The information on the Cloak states

If you are attacked, you can use the cloak three times per day to produce one or the other of the following effects. You can gain concealment for 1 round, or you can teleport to a space you can see clearly up to 10 feet in any direction.

So it only works against an action defined as an attack. As such you need to decide what exactly a grapple is considered to be.

http://www.d20srd.org/srd/combat/actionsInCombat.htm

Has Grapples listed under its miscellaneous actions Table near the bottom of the page.

The note connected to it says

2. These attack forms substitute for a melee attack, not an action. As melee attacks, they can be used once in an attack or charge action, one or more times in a full attack action, or even as an attack of opportunity.

So Grapples according to the SRD are defined as Melee Attack Actions. As such...

The Shadow Cloak would allow you to escape from a grapple attempt. It cannot however be used to escape from within an already successfully established Grapple.

Generally¹, the Polymorphed creature can't grapple

My reading on the other action is other than spells. Grapple requires a free hand, thus it obviously requires hands.

The target of your grapple must be no more than one s/ize larger than you and must be within your reach. Using at least one free hand, you try to seize the target by making a grapple check instead of an Attack roll

This reading means polymorphed creatures can not take the Grapple action. (Yes, I'm aware Grapple is not an action - it's just a special Attack action, but I think this way is easier to understand.)

There are exceptions

Monsters don't (necessarily) follow usual Grapple rules, as stated in the MM p. 11.

Many monsters have special² attacks that allow them to quickly grapple prey. When a monster hits with such an attack, it doesn't need to make an additional ability check to determine whether the grapple succeeds, unless the attack says otherwise. A creature grappled by the monster can use its action to try to escape. To do so, it must succeed on a Strength (Athletics) or Dexterity (Acrobatics) check against the escape DC in the monster's stat block. If no escape DC is given, assume the DC is 10 +the monster's Strength (Athletics) modifier.

This leads to some monsters being able to grapple creatures without actually requiring hands. For example, a Giant Octupus has the action

Tentacles. Melee Weapon Attack: +5 to hit, reach 15 ft., one target. Hit: 10 (2d6 + 3) bludgeoning damage. If the target is a creature, it is grappled (escape DC 16).

This action does not require hands, thus it is allowed. The action states that the creature is grappled, so the creature is grappled. Thus, Giant Octopus can grapple normally through the use of Tentacles.

For other examples, T-rex and Giant Toad also have actions that grapple using Bite, which is more far away from "hands" than the tentacles mentioned above.

Crawford has also spoken on these matters:

Q: Polymorph says I cant take actions that use hands. Grapple action requires a free hand. Can Polymorphed PC use grapple?

A: I would allow it if the beast form had an appendage or a bite that could potentially restrain someone.

Note that this doesn't follow the usual "These are the official rules" style of writing from Crawford - it just says "I would allow it", making unclear if that's how it actually works. This is an old tweet, though, and that might be the reason for it. Also there is a lack of "allow only if", but I guess that's implicit easily.

Possible house rules

Personally, I don't see any (balance/mechanical) problems with allowing a Giant Ape to grapple normally. I would think the restriction of using hands was more for locks, doors and other things requiring Dex than actual combat actions. My reading on RAW and Crawford's tweets say this isn't the official rule, though.

¹ Here, generally means that it's the rule that holds true unless stated otherwise, e.g., unless the beast has a specific action that lets it grapple without hands, as described in the following section. The fact that this is the general rule is the reason beasts are required to have the specific action to be able to grapple as we would usually imagine, for example, a Boa Constrictor doing.

² Since the "general vs specific" discussion is constant, I'd like to emphasize that the Monster Manual itself says these attacks are special, i.e., not the general/usual rule. These are exceptional, special, specific attacks described in their statblocks that apply exclusively to the monster in that statblock, thus it can't be general in any way. Even if many monsters have these special attacks, all of them are specific to each monster, and the general rule is still applied to all of them.