im not sure what you know about electric wheel chairs and arduinos so I will just post the basics
The dc motors on a wheel chair are basic dc motors with built in electrical brake.
The brake is power to release so when the wheel chair is off the brake on the back of the motor is applied.
The normal controls are speed limiting, on/off, battery monitor and joystick steering.
The Ardunio can output a straight dc output or a pwm (pulse width modulated) output. To control the motors you will need to input the variable signal via analog input from the joystick then output a pwm signal to the H bridge drives to control the motors. Another input from a pot (variable resistor) should be read on a analog pin which will be interpreted by the arduino to act as a speed cap for the motors (it will reduce the max pwm output).
If you are using proper wheel chair motors you will need a output for the brakes which will require a mosfet or a relay as the arduino can not supply the power required to release the brakes.
Wheel chair used motors are cheap in America at about $50 to $75 including gearbox and wheel for a set. The original controls for a wheel chair are complicated as the source code and program is not open source. (the controllers do fail which is why you can buy cheap motors). Most run on 24vdc with a locked motor current of around 40 amps. A smaller motor with chain drive will work but they lack a brake so the operator will have to be careful when they get out of the chair as it may roll backwards.