I want to second the caution about driving a four wheel vehicle. You can get in bad trouble quickly in one where the front wheels can turn into the body of the vehicle. When that happens, the whole vehicle will twist up on itself and can flip over. It is bad enough when you are just making a regular turn, going forward, as even then the vehicle can twist and flip. But, it is very dangerous if the horse is refusing to go forward, as even a few backward steps will put the wheels into the body of the vehicle and cause a problem.
It is my understanding that there is a difference between a 'cut-under' and a 'fifth wheel' vehicle. A cut-under will give you some extra room as the wheels can come under the vehicle to some degree, but with a cut-under, at some point they will contact some part of the vehicle, either the 'box' or the undercarriage (frame), so can stil lock the wheels. It isn't as easy to do, so is safer than a vehicle that is not cut under.
A fifth-wheel vehicle pivots completely, without the wheels ever able to contact the box or the undercarriage. You can turn as sharp as you want going forward, the vehicle will just pivot around. If the horse goes backward, it can come around until it gets next to the vehicle box and can't come around any more, but it won't twist the vehicle to cause it to flip. So, I think a fifth wheel is safer than even a cut-under. You can get the fifth-wheel type design in both a 'buggy' (light weight vehicle) or a 'wagon' (farm or 'Budweiser' style heavier vehicle).
Generally, buggies use the same type of harness you would use on a cart, and you use a 'heavy' harness, like a work horse would use, for a wagon. Of course, either one, as long as it is strong enough and fits the horse well, will work to actually pull either type of vehicle.
