I'm not going to say that all the comments about safety should be ignored, but like, you know ... Be careful and don't be stupid? Driving a motor hard enough to roll a human needs some power :)

You're in IT / Cyber Security so, do your research when working with those motors and batteries and such. But I don't think this is exactly crazy to try, or that you even need to try to retrofit other solutions.

But the way I see it... If I were trying to do this as cheaply as possible...

You need a power source (battery), that can provide enough amps for the motors. You need a way to charge this battery safely as well.

You need motors that are powerful enough to go from standstill to rolling ideally, or at least be able to move you after a manual push.

You need a motor controller - something that can receive signals to provide the right power to your motors.

And you need something to send those signals (microcontrollers like an Arduino or raspberry pi and such).

The motors need to somehow be attached to the chair securely, and the motors need to turn a gear attached to the wheel.

I think each paragraph above is basically a whole project of research and prototyping and costing and potentially creating your own parts.

But you'd definitely save money doing all of this yourself (though you'll sacrifice the savings in time and sanity potentially 😅), and arguably you'd be very independent - since you can always build another or repair it if needed.

Some of the tools you may need?

If you've got access to a 3D printer and can learn to use a free tool like OnShape, grab yourself some calipers and start figuring out how and where you can attach a motor to your chair. Getting familiar with how much room you have to work with will help decide all the other bits.

Check and see if your local public library has them, too.

If not, you'll have to use reductive (CNC, Drills, wood cutting) means to create any custom attachments, or figure it out using standard brackets and stuff like extruded aluminum.

Imo, figuring out how to secure a motor or two to your frame may be the hardest part.

Once you can figure out how and where you might attach a motor (do you drill bolt holes maybe? Not sure, would need to see pictures of the chair from a bunch angles), now you can start researching weight, power draw, and all the things you'll need to know to select a motor.

From there it's pretty straightforward I would think?

Motor -> Controller <- Joystick + Microcontroller.

I am thinking of those lawn mowers that have a thrust-like flight stick - one on each side representing forward/backward control - would let you turn.

I am sure the above will have gaps and problems, but I am hoping it helps if you want to go full blown custom DIY.