As long as both motors only run in one direction, the basic idea is pretty simple. Joren Vaes has priority. However, his answer can be expanded.
First, let's say you're willing to drive your motors in bang-bang mode (either on or off). Then
simulate this circuit – Schematic created using CircuitLab
will do the job, with a few reservations.
1) The motor will not start to turn on until the joystick is pretty far from the zero position, so there will be a large dead zone which will make precision operation difficult.
2) The motor will go from full off to full on over a fairly narrow deflection range. This is actually a good thing, since
3) If you try for partly on and succeed, the MOSFET (M1) will start to get very hot very quickly unless you provide a good heatsink for it. Like, it will die. And if the motor draws multiple amps of current (and a mobility scooter will indeed do that) you will need a much bigger heat sink than you think. Just a warning. On the other hand, if you just use the controller to drive the motors full on or full off, you won't need much heatsink.
Also note that I'm assuming you use a single 12 volt battery. This is important. If you use a higher-voltage setup, like 2 12-volt batteries in series, you will produce 24 volts, and you will be able to kill the MOSFET by applying too much gate voltage. If that's a problem, start another question.
Finally, keep in mind that, as I stated at the beginning, this will only work if you are willing to drive your motors in one direction. There is no simple way to allow reverse drive for this sort of setup.
EDIT - And you should also keep in mind that driving one motor at a time is unlikely to do anything you want to do. With this setup, the vehicle will spin in place (approximately) and your only choice is which direction it spins.