Vehicles/mounts create such an advantage for those who have them that their use would never have died out.....if horses have gone extinct (a point of lore I find objectionable), then someone, somewhere, would have found an alternative. For that matter, the folks at the Institute have the intellectual firepower to create that alternative. I hope there are vehicles and/or mounts in FO4. A horse (or equivalent) or a dirt bike would probably be the best as they can operate off road. But no transport but the Mk I Mod 0 Foot is an artificial constraint whose time has come and gone, IMO.
The Boomers used bio-fuel from crops they grew to fuel their B-29...a faction controlling a decent amount of arable land could make a lot of money selling it. They could also run on Fission Batteries, or even refurbished Pre-War nuclear engines (hopefully they won't explode so readily).