A player needs to hide the fact they are speaking, have the puppet act in a crowd pleasing manner, and fool children into believing the puppet is alive/the source of the sound.
How do I handle this mixture of stealth, performance, and deception? Do multi-skill checks exist?
It would be the Performance skill since you are trying to perform for a crowd.
Deception wouldn't really be applicable since you aren't really trying to convince someone of something that isn't true using words. ie. You are deceiving people, but not by using speaking lies to them.
Sleight of hand doesn't cover it since you aren't using your hands to make the deception per se.
Stealth doesn't cover it since you aren't trying to hide yourself from sight.
A DM could call for multiple skill checks of different types. For example, if you are trying to run through an obstacle course you could use both Athletics and Acrobatics. It is up to them on how to deal with the different roll results.
You're really asking two very distinct questions, so we'll tackle them one at a time.
We'll start with
Sometimes the skill work in sequence; you need to successfully perform one check in order for the opportunity to perform the second. For instance a caber toss. You would first do an Athletics check, to see if you can even lift the caber. Then you would do a second Athletics to see if you can hurl it correctly (end over end). Or maybe you're trying to pick pocket someone. First you would do a Performance check to get their attention and be your "volunteer", and then a Slight-of-hand check to see if you can lift their purse while distracted.
More often, at least at my table, they are synergistic; one helps another. An example I gave in another question is fishing. You would roll a Survival check to fish or stalk wildlife. But if you did a successful nature check first, I would give advantage (or a better result) to the Survival check. Why? Because Survival says you can hunt, but if you're familiar with the environment (Nature) you know more about your prey. Like what bait to use, the best places to look for a certain plant, and so on. Another example would be bull fighting. Use an Athletics check to dodge a charging bull, but a successful Animal Handling would give you advantage as you can "read" the bull better and be prepared.
Another example would be putting on a show for the Clerics of Northhaven. Sure, it's a Performance check. But if you didn't know that their religion forbids wind instruments in their temples, you could be in for a nasty reception! Good thing you did well on the Religion check.
There are also cases where its an either/or. When you're climbing a tree, you can either Athletics up (bear hug and shimmy up) or Acrobatics up (leap from branch to branch). And they do not need to save the same DC! One can be easier/harder than the other.
Given that
However, I would give advantage to that roll if they first did a successful Constitution check. Why Constitution?
But per the Basic Rules:
Constitution checks are uncommon, and no skills apply to Constitution checks, because the endurance this ability represents is largely passive rather than involving a specific effort on the part of a character or monster. A Constitution check can model your attempt to push beyond normal limits, however.
You are trying to resist the temptation to move your lips, to hide your breathing, and be as still as possible while making a puppet come alive.
Or just use Thaumaturgy and Minor Illusion and be done with it.
