So we essentially have a robot chassis and drive system that can carry a person! Our first idea was to make it remote controlled.
Please read this carefully - your first order of business is -not- to make it remote controlled.
What you need to first do is plan on how to control the machine safely
. This isn't going to be some small robot on a desktop or floor that you can easily pick up or restrain with your hands. This is ultimately a device with a lot of power, with some weight (you don't mention what kind of "wheelchair" it is - so it is difficult to picture, but it likely weighs near 100 pounds, if not more), and it can probably get up to at least walking speed, if not a bit faster.
Such a machine can be dangerous if it gets out of control.
So, your first order of business is to plan the programming, control scheme, and emergency stop system that will be used to prevent such a powered chair from running amok and damaging property and person - well before you even think about connecting the Arduino to it.
You should first attempt to locate specifications and data-sheets for the controller used by the wheelchair, including the controller interface for the joystick (or whatever controls it). Depending on how things are set up, you might be looking at a PWM interface, or some kind of voltage-controlled interface (with the joystick acting in as a combined dual-axis voltage divider). If you look around on this forum, I know I have posted about this before, with some links, for another individual planning on building something similar.
Don't think about bypassing the controller; it is unlikely that you could build your own controller cheaper or more reliable, especially for the voltage and current levels we are talking about (the L298 will not cut it here - think paralleled MOSFETs, most likely). There are companies that manufacture controllers for such systems; if you had the specs for the motor or the chair (voltage and current draw, stall current, etc) - they will help to determine what kind of controller to buy, should you go that route. Be prepared that such a controller will not be cheap; for a wheelchair, if the controller isn't costing more than $100.00, it is probably undersized. You might also look into "combat robotics" controllers - these typically have inputs for emergency stop functions, which you are going to want.
Getting that emergency stop function built in can be tricky - probably the best way (if the existing controller, or a purchased controller, doesn't already have one) would be to use a vehicle starter relay to cut the power to the motors and/or controller (and possibly the entire system). Either that, or a 24 VDC contactor relay. Whatever method you decide, you'll want to be able to control it via three methods concurrently: 1) digital I/O from the Arduino or whatever other digital system you use, 2) an R/C interface - a cheap "auto-start" key-fob transmitter controller could work here, and 3) a physical "pull-out" dead-man's switch with a large, easily graspable handle mounted on the robot.
Once you have that sorted out, you are going to want to do all of your development of the control system, as much as possible, away from the chair. Develop a testing harness if you have to, running motors or LEDs or whatever (this is something you'll have to determine), such that you can code and see what the system should do - long before you hook it up to the wheelchair controller, and the motors themselves. This kind of setup serves a couple of purposes: It makes it safe for you to develop and test untested control code, and it makes it easy to develop without having to be near the chair and its unwieldiness (plus, with a duplicate set of test harnesses, you could each do development at your respective homes, without explicitly needing the chair nearby).
Once you get to a point where you think you have something working, you can then move to hook it up to the chair - but you should first prep the chair in this phase of testing by removing the wheels and mounting the chair and control system such that the motors don't snag on anything. Remove and/or tie up all long sleeve and loose clothing. No dangling jewelry allowed! Tie back any long hair, too. Finally, once you have the system set up and running, resist any urge to touch spinning parts. These aren't small motors, they are powerful gear motors that can, if given a chance, cause grievous injury (you do want to keep all of your fingers, right?).
When you feel solid about this testing phase, having tested all control systems for proper functioning, as well as failsafes and emergency stop systems (by the way, you will want to incorporate a method to trigger the e-stop system should the Arduino or whatever else is in the control chain stops responding to the controlling PC, radio system, etc - should a wire come loose, a cord damaged, radio interference, bad control signals, over-currents, noise, etc), then you can move to putting the wheels back on. You should then do a final stage of testing "up on blocks", not on the ground - until you are confident that the system is fully responding the way you intend it to.
By the way - you might want to figure in some form of sensor system up-front as part of the e-stop system; IR, ultrasonics, physical bump sensors, etc - all should be thought of in the design up front as a part of that early planning stage. If you decide to make this into a real robotic system, you'll want and need them anyway, so put on a few, at least on the front, in the beginning.
Now, you might think I am taking all the fun out of a project like this; this is not my intent at all. My intent is to hopefully educate you on how to stay safe and sound during a project like this. There is a ton of work to do to make this a reality, and plenty of fun to be had while staying safe. The last thing you want during any project is to find yourself on a visit to the emergency room because you decided to take shortcuts or ignore safeguards.
Have fun, and enjoy the project...
/BTW - your friend's parents might want to make dead certain that the old chair is theirs to keep; if not, and you rip into it - it won't be a cheap purchase for them to reimburse...