No (Probably).
Cheek pouches can only hold 1 cubic foot. Since 1 foot is the minimum length/height a Tiny creature can be (Table 8-1, p. 256 Starfinder Core Rulebook), it's likely the drone is too big.
Then you have to consider:
Ysoki have two cheek pouches with a combined volume of 1 cubic foot. Since the volume is divided between these pouches, it's extremely unlikely a ysoki can use up their cheek pouch volume with a single large object. Cheek pouches are designed to hold many small objects, not one really large object.
The drone would need to fit in the ysoki's mouth. Even with cheek pouches, that's extremely unlikely.
You might be able to convince your GM that your drone is compact enough to fit, but then you'd have to convince them that such a compact little drone is sturdy enough to hold weapons, armor, etc.