General terms:
* Runner
* Rusher
* Back
* Offensive player
* Ball handler
More specific terms:
* Running back (American Football)
* Halfback (American Football)
* Fullback (American Football)
* Wingback (Rugby)
* Centre (Rugby)
* Forward (Rugby)
Other possibilities:
* Holder
* Possessor
* Bearer
The best choice will depend on the specific situation and sport.