The 28-year-old former Kidderminster Harriers man picked up the injury during the first half of Tuesday night's 3-1 home victory over Cambridge United.
He was replaced by Durrell Berry at half-time and will now be forced to sit out the remaining three matches in the Robins' battle to survive.
Vaughan has made 35 appearances this term since arriving from Aggborough on a two year deal last summer.