Could you give us some context (for example, the question to which your sentence is a reply)?
The reason I ask is that you can start a sentence with maybe and have different meanings, depending on whether there is a comma.
Your context helps a lot!
It makes clear that there should be no comma after Maybe.
We would not say Maybe they might but either Maybe they kept the food in the fridge (or They might have kept the food in the fridge).
An example of maybe with a pause (better to avoid a comma and have a dash (--) or two sentences.
A "Shall we have a look in the fridge?"
B "Maybe. The food could be there."
The word maybe here (=Perhaps) refers to the whole sentence spoken by A.
It's tautology -- saying the same thing twice. (This does not automatically make it wrong.)
But if you wanted, it is possible to combine maybe with could, e.g. "Maybe we could have a look in the fridge?"
Using perhaps you could use miight: "Perhaps they might/could have left the food in the fridge."
My feeling about maybe + might is just a feeling. It is not based on any evidence. But I would recommend you to use either maybe or might, not both together.