Definitely your own, you know what works for you and you know what product to use in each individual situation. Customers might have a lot of crap that is no good and each time you use their products you have to spend additional time finding them and reading about what they are for.
I would highly suggest to get your own, even in the begingng. You don't want to ask your client where their supplies are. And what if they aren't there when you are cleaning?
Not having all the supplies by yourself just screams unprofessional to protential clients. So invest a few bucks and buy all of your own supplies.