Most people think that getting a warthog in the shaft is impossible. But it really isn't. All it takes is a few hogs.
Get a warthog and bring it into the structure. Then ram it through the doorway and it should be halfway through. Then back up a bit and exit so you'll get out. Then grab one of the other warthogs and ram it. It should go through.
You can get all 3 warthogs in the shaft, but I'm not sure how to get the 3rd one in there.
You're not going to be able to make it very far riding, though, Unless you ride down to the lower ledges (or the Silent Cartographer room ledge). Just adjust the hog so its facing them and ride. However, the hog will likely flip over and you'll probably receive some damage. So over shield is advised when doing this.
You can also use the warthog to get far down in the shaft (and the bottom). You just have to ride from the 'Shafted' cut scene ledge into that lit room down to the left. Or onto the little ledges in the middle. But try to get into the room, as you'll have more room to drive off of.