I do the same thing here in the U.K. I got into it after my mother died as I realised that in the last 2 years of her life she would have been very isolated if it weren't for me keeping her company every day.
There is a company, who I think is in Canada, who started up the same business and is now selling Franchises (I am new here but sure you can't post a web address so think of the name of a film with Jessica Tandy who employed a driver and her onscreen name was Daisy, lol)
I am pretty sure that U.K law and U.S will be the same in that you will be acting as a companion/chauffeur(sp?) so won't need a taxi licence but it would be advisable to take out insurance in case of an accident.
The elderly need help in so many ways but most of all they need companionship. Make up flyers and cards offering accompanied trips to the shops, doctors, hairdresser, dentist etc. While you are with them ask about their ancestors and offer to take them to the cemetary where they are buried (maybe sounds ghoulish but old folk think of their deceased relatives often). One of my clients insist on paying £45 (around $70) each time I take them out as they enjoy my company.
Well worth thinking about as they will also ask you to do other odd jobs for them