Nothing is impossible.
I would look at using parts from one of these.
Also have a look at this site:
http://mobile.robotmarketplace.com
For steering you might wanna consider individual motors at the drive wheels and swivel wheels at the front, like on a shopping trolley (and most battle robots) rather than a steering rack.
Possibly even 6 wheels, with swivels at either end and drive wheels in the middle.
Or drive wheels in the middle and a single swivel wheel at either end.
