Well, social skills are usually based on charisma.
Could we say for example:
- use powers to operate illithid device = int
- use power to gain understanding of something or learn the intentions of someone = wis
- use powers to manipulate the behaviour of someone = cha
- causing a physical effect is based on a physical stat and so on
So the tadpole gives you powers.
But you chose to use these powers (or not using them) with your own body and mind.
So the stat used depends on the exact effect you want to achieve with those powers.
I don't think that using the tadpole should require any physical stats. Nothing you do with it involves physical effort - physical strength, nimbleness, or heartiness.
Wisdom makes sense as you're feeling your way through using the tadpole, and often using it to gain insights into other's brains (connecting your tadpole to theirs mentally).
Charisma could make sense as you're forcing your will onto either your tadpole or onto the target's tadpole, using that connection overwhelming the target's mental defenses with your force of personality.
Intelligence could make sense for very limited checks like operating an ilithid device that operates on machine logic.
