Replace a pot with an arduino

I have an old wheelchair controller and I am looking at replacing the joystick with and arduino the voltages across the pots in the joystick is 5v would it be possible to simply connect an analog output to the wire currently connected to the centre of each pot in the joystick and output 0 - 5v?

You'd have to use a DAC of some sort; arduino output isn't analog. That would -probably- work though, just be careful there's no feedback that might damage the arduino.

If the pots are operating as simple adjustable voltage dividers then yes, you could possibly use PWM and a low pass filter to create an analogue voltage to feed to the "wiper" connection. Be sure to remove the pots, or they will bias the analogue voltage levels and screw up your results.

A better way would be to replace the pots with a digital potentiometer chip (there are many of them available from many manufacturers) which you could control via I²C or SPI. This would remove any problems of the impedance of the target device messing up the low pass filter on the PWM. Yes, you could buffer the filtered PWM through an op-amp buffer, but it'd be simpler to just use a digipot.

I believe there is an example sketch in the "Examples" within the Arduino IDE for controlling a digital potentiometer.