Some days I wish I had an errand runner... I think it is a great idea and that elderly people would especially be into this, but it might be a little bit challenging.
I would make sure to limit where you go if I were you I would keep it to stores in my town, grocery store, post office, Walmart, the drug store etc. and charge one flat rate for gas like 10.00 a trip, then charge by the half hour for your service.
Where I think it would get hard is grocery shopping for someone else
if you did not get the brand they wanted, or if they did not give you enough money, and the money part would be hard too..... I wish you luck!