Contrary to what I guessed initially, I now think the question has a great answer: the functor is representable if and only if $M$ is locally free, and the proof is EGA I, 9.4.10.
Edit: this is an answer to an earlier version of the question.
Contrary to what I guessed initially, I now think the question has a great answer: the functor is representable if and only if $M$ is locally free, and the proof is EGA I, 9.4.10.
Edit: this is an answer to an earlier version of the question.