They can’t ask for proof, but even if they could, there’s no proof to be had because there’s no kind of universal service animal license.

They also can’t charge you a pet deposit or pet rent because it’s a service animal, not a pet.

They can only ask you two questions:

1. is the dog a service animal required because of a disability
2. what work or task has the dog been trained to perform.

That’s it. Print the pamphlet, and send it to them

https://disability.mo.gov/files/ServiceAnimalsAllowed.pdf