Well - I've been doing it for years, and I don't think it's unprofessional! How else (reasonably) are you going to get water? I don't think for a minute your clients expect you to carry warm water IN ADDITION to everything else you have to schlep around.
I use a clients' sink for filling my foot spa and manicure bowl, but I bring my own towels (although sometimes they have their own sitting out, waiting for me). One of my clients has a really slow boiler, so she even boils the kettle for me in advance so I don't have to use tons of water waiting for it to heat up.
If you are professional in all other aspects - impeccable hygiene and safety procedures, on time, and providing excellent service - no one will bat an eye if you ask them for water. But if you are lacking in any of those areas, the water is the least of your worries.
Mobile work is hard but it's a lot of fun and your clients will be appreciative and most will go out of their way to make your job easier.
Have fun!