No, you can't fool the spell, you have to fool the target another way
The spell Sending specifies that recognizing the sender is a part of the spell's function, and nothing in Alter Self would change that, as it is appearance based.
Sending (as you quoted yourself):
The creature hears the message in its mind, recognizes you as the sender if it knows you, and can answer in a like manner immediately
So if the target knows you, not just your appearance, then it will know you sent the Sending. What you look like at the time is irrelevant, the spell functions as it says it does.
Alter Self:
Change Appearance. You transform your appearance. You decide what you look like, including your height, weight, facial features, sound of your voice, hair length, coloration and distinguishing characteristics, if any. You can make yourself appear as a member of another race, though none of your statistics change. You also can't appear as a creature of a different size than you, and your basic shape stays the same; if you're bipedal, you can't use this spell to become quadrupedal, for instance. At any time for the duration of the spell, you can use your action to change your appearance in this way again.
Nothing here says anything about disguising you in any other way besides visually. It does say the sound of your voice changes, but Sending makes the target hear the message in their mind, not with their ears, so changing your voice doesn't affect that.
Now, by what you have described, you'd have to have used a dispatcher who does not know you to begin with and to have already disguised yourself before speaking to them, so when the Sending gets back to them, they will know it comes from the same person they sent out. Basically, you have to fool the dispatcher, not the Sending spell.
To be clear, if your name is Bob the Baker and you have an alter-ego as Skinny Larry, and you cast sending to someone who only knows you as Skinny Larry, Sending will not out you as Bob the Baker, they'll know the Sending came from the 'you' that they were introduced to.
Using the general guideline that "spells do what they say they do", I would rule that the familiar can indeed answer. Find familiar says that you can "communicate with [your familiar] telepathically", but doesn't otherwise grant you any special ability to speak to the creature, which "has the statistics of the chosen form" of animal, none of which know any languages.
Since sending specifically says that "creatures with an Intelligence score of at least 1" can understand you, with no special requirement that they speak a language, and that "the creature ... can answer in a like manner", I would rule that the spell magically allows the familiar (or any creature known to you with an Intelligence of 1 or higher) to both understand your message and to reply, to the best of its ability (i.e. its animal level of understanding would be translated into up to 25 words by your DM).
This seems very reasonable for the power level expected of level 3 spells; speak with plants, by comparison, makes all plants within 30 feet capable of communicating with the caster for 10 minutes, and (it can be inferred from the terrain-altering effects) compels them to follow your simple commands, so a single 25-word exchange across space and even dimensions with a creature with an Intelligence score seems fair.
Best Answer
Ring of Spell Storing
Have the NPC use a Ring of Spell Storing, a "standard" magic item. It can store up to 5 levels of spell. Some other relevant features for the item:
So the NPC could have found such a ring with a single Sending in it at some point. They had it identified, and now's time to use that Sending. No wizard needed at this time.
The NPC could also have had the ring for a long time already (as a family heirloom maybe), and they just need to free 3 levels worth of room from the ring, and find a Wizard to cast a single sending to the ring. Or maybe even in this case the ring has Sending already, as it seems a very logical thing to put into a ring, for emergency use.