Can your mother stand up? If she can, pull the wheelchair up to the side of the car of the passenger front seat. Make sure the door is already open
and that the wheelchair brakes are on.
Help your mother to stand up, step forward and then turn so that her back is to the passenger seat. Have her sit down with her feet still on the ground. Put your left hand under her knees and your right hand under her ankles and pivot her legs around so that she is facing forward looking out the front window of the car. You can then pull out the seat belt, leaning over her to fasten it.
The next task is to fold up the wheelchair and put it in the trunk of the car.
This can be very tiring if you have to do it too often. Just going to a doctor's appointment means you have to do this 4 times.
A sliding board can be a help. There are safety issues so you should get trained by a physical therapist, physiotherapist, or even a wheechair expert. Many medical equipment store have people who have become experts about wheelchairs and how to use them.
You might want to consider a wheelchair transportation servicve. Many cities have services with vans that come to your door. The wheelchair can be picked up with the patient in it. The elder can sit in the wheelchair on the way to the appointment. You would meet your mother there, greeting her as she gets out of the van. This is safer and less exhausting than trying to make too many transfers.
Also, helping a grandparent into a car can be a way to get young people to help elders. Again, training is key. Make sure that any person who will be helping to transfer someone from wheelchair to auto, is properly trained to avoid back injury..