All depends. Depends where you live and what resources are available. Where I live, no. There is no free transportation for grocery shopping and banking. But sometimes you can get a discounted monthly or annual bus pass. Most people have to rely on friends and family for transportation if they do not have a car.
Locally one grocery store offers delivery to shut ins, a local pharmacy does too. Handy Dart is low cost public transportation that can be booked for appointments. Dad qualifies for up to $200 in taxi rides each month but only has to pay 1/2.
In moms area the public transportation company has a program for "disabled" including elderly who may not be "officially" disabled. Anyone who cant get to, on and off the bus on their own really can qualify. They provide van rides anywhere in town and for medical appointments to surrounding towns even. It isn't free but it's less than bus fare even when you by ticket books. Check with the local public transportation company and or Agency on Aging to see what's available.
Call your local Office of Aging. In my area they have small buses that take disabled and Seniors to appts and shopping. NJ transit has buses that go from town to town. Both have small fees. If on Medicaid for health insurance they have transportation.
Most banking can be done online.