I thought there was a car share website in Leeds - I may be mistaken - I'm sure there are organisations that specialise in such matters.
Just be careful. It sounds a good idea - but can be a pain if they are ill one day and you're left on the street corner waiting for your lift. I think you'd be better off on the train. It's easier to blame the train for being let than someone else !
In fact just looked and found this link
http://www.liftsharesolutions.com/?gclid=CLfuq eOFyYgCFThhMAodyE_VBw
or this one:
http://www.lowimpact.org/linkscarsharing.htm
Google : Car Sharing Leeds to York and sift through them.