I've worked in the personal mobility field (wheelchairs and mobility scooters) in the past and here a few things to keep in mind.
As Greg Locock said, do you really need petrol? and if you are going petrol why bother with the electric side of things?
From the electric side of things electric scooters use 2 x 12v deep cycle batteries in series to give 24v. the controllers can handle from 70 to 160Amps for transient loads (Acceleration, kerb climbing etc.) If you use 50% as an efficiency for the motor/gearbox thats up to 2hp power output. The acceleration some scooters are capable of when the programming is wound up is impressive. Scooters with a range of up to 35miles are available.
Electric motors each wheel will require some way of varying the speeds individually with respect to steering input if you want any real success at turning corners.
Single motor on axle will require a differential of some description for the same reason. Will also lose all regenerative braking if an open diff is used and a wheel leaves the ground (usually when trying to avoid a collision).
Braking, mechanical or regenerative braking, or both? (electric motor is used as a generator to recharge batteries while also provide braking force). Common on bigger/faster mobility scooters is a mountain bike brake disc and caliper on the rear axle as a secondary/backup system. How good is your mothers hand strength?
Mountain bike shock absorbers are also used on scooters to provide suspension. Usually an electric scooter has the motor and batteries mounted in a rear sub frame that pivots at the floor pan level just behind the seat.
Safety. What happens if the throttle sticks open? Also limit maximum speed to something reasonable. Take into consideration your mothers eyesight and reaction time. We had to limit the speed of some of the scooters we sold because we had 80yo men with poor eyesight terrorising the malls and retirement villages at ~10mph in a scooter that weighed in at over 250 kg (including passenger and basket full of shopping)
Mounting and dismounting. A seat that swivels 90+ degrees will be a very probably be a desirable thing for a woman that has had knee surgery and may still be using crutches, just make sure it latches in the forward facing position while she is driving.
Axle loading. Need to keep most of the weight on the drive wheels to maintain enough traction to be able to drive up small obstacles (kerbs, doorways etc). Keep enough weight on the front axle to be able to steer on smooth/slippery surfaces however. If you build a large outdoor only type scooter with large diameter tyres this wont be such a big issue.
I would suggest modifying an existing mobility scooter chassis if you can find one in an appropriate size/price. It will have steering linkages, tiltable steering column etc already setup and save a lot of fabrication/design time.
Square section steel tube is the most common construction material for the frames.
Hope thats of some help.