Great answers above. She can apply for county transportation (if you are in the US & live in a relatively urban area) through the local department of Aging. The fee is sliding scale according to income and like the poster stated above, a couple of bucks each way.
Sorry, I don't know of any free service. I sure could have used one for my very elderly parents [90's] once my Dad stopped driving, as they had a ton of doctor appointments. I used up all the vacation days [couple hours here, couple hours there].... even used up all the sick days... and had some days without pay.
Check with your County Agency on Aging to see if they have a mini-bus service that could pick your Mom up and take her to the doctors, and return later on to bring her back home [she might have to sit and wait for the bus to come back]. This service is usually a couple dollars each way.
You can try calling a professional caregiving Agency. If the appointments are planned far enough ahead, the Agency can arrange to have one of their caregivers to take your Mom to her appointment and wait for her. Note that the cost could be around $20-$30 per hour, with a 2 or 3 hour minimum [depends on location and if the caregiver has a long drive, mileage will also be charged to the bill].
Your Mom is still quite young. What type of medical issues does she have that is limiting her so she can't go on her own?
Check with your County Agency on Aging to see if they have a mini-bus service that could pick your Mom up and take her to the doctors, and return later on to bring her back home [she might have to sit and wait for the bus to come back]. This service is usually a couple dollars each way.
Your Mom is still quite young. What type of medical issues does she have that is limiting her so she can't go on her own?