I would consider reporting them to the Health Dept. for not providing soap in the bathrooms. This is contrary to public health requirements and standards that business are to maintain. At least discuss it with them, this is not an acceptable situation and if it requires the Health Dept. to force their hands then so be it.
In the meantime use a bathroom that has soap even if that means going way out of your way to do so. I would email my boss and let them know the location staff is having to travel to wash their hands, so they can't claim they weren't aware. Whatever you do, do not buy your own supplies, this is not your responsibility and just rewards their cheapness.
Save every note, interaction and email you send for future reference, forward email to another mailbox so they can't disappear. You may need this information in the future, hopefully you won't. Eliminate every possibility that gives them "plausible deniability".
Every time you need a paperclip, pen, tape, printer paper or staple go to HR and request them.