Hi Marj,
For my mobile nail business I used a portable manicure table which folded into a bag I could carry on my shoulder, and a large beauty box on wheels which housed all my product/tools/towels/client cards/money tin etc. I had a small car at the time and these fitted in no problem. I also always made sure I had a long extension lead with me and used this nearly every time.
With regards to chairs, I used chairs at the clients' house. Usually when a client would book they would ask me if they need to prepare anything, and I would say they only need to supply 2 chairs. I never had a problem where chairs were concerned.
Also remember you need to arrange Public Liability Insurance - I got mine from BABTAC and it was only about £50 per year.
Hope that helps.
Laura