Do daycares usually let you use cloth diapers?

It would depend on the laws of your state and possibly the rules of the daycare's accrediting agency. I had one who was allergic to diapers and had to be in cloth, so when we checked the rules for my daycare, they were allowed to use them. You just had to either use a cover, or just use a modern diaper. If you wanted to use the really old fashioned ones with the safety pins, you had to use the old fashioned plastic pants. Any of the modern cloth diapers were fine. Obviously, I had to bring the diaper bags for them to put the diapers in, and they couldn't do any clean-up or anything.

No, unless you go to a private home daycare where the owner doesnt mind doing it, but at a normal daycare, they use disposable, otherwise they have to deal with washing the diapers and if you just have they lay the diapers aside and send them home with you to wash, then they have to deal with sanitary storage of poopy diapers.

To be honest all the daycares I went and looked at supplied their own nappies for convenience. I don't think most daycares would be flexible and allow this but you never know you might find one that will? Just call around and ask, then go in and have a tour of any you find that you like.

My oldest daughter went to day care and they required disposable diapers and didn't make exceptions.

Edited: I live in the US. We have to supply the diapers that the day care uses on our children.