It seemed like a good idea to start with the simplest thing, so I designed an oxcart. It took a week of fumbling around. I think the main problem besides acute beginneritis stemmed from using projections as a guide. Things improved once I switched to Cartesian products.
The body of the 4D oxcart is a hyperprism. It has a top, bottom, and six sides. There are four wheels, one each for the ana, kata, left, and right sides. It is steered by a bar in the front that pivots only up and down.
Perhaps I spent most of the time worrying about the axle upon which the hyperprism rests. One axle or two? What shape should those axle(s) be? Should it be a one axle then it is the Cartesian product of a planar object -- a square would do -- with a small disk. I think that would be all of heavy, weak, and difficult to make, so I went for two perpendicular axles. A trouble I had was trying to use as an axle a spherinder, the Cartesian product of a small sphere and a line. Such has many possible planes of rotation is so is not a stable mount for a wheel. It is better to have as an axle a cubinder, the Cartesian product of a disk with a long skinny rectangle. Then there is only one possible plane of rotation. (The axle doesn't rotate, but it must be possible for a hub to rotate around a portion of it.)
The two axles could be merged into one to make an X. But I think that would be difficult to make and less durable to boot. Better to have one axle slightly closer to the front of the cart than the other, so that the two axles will be at the same height but not intersect.
The hub of the wheel is the Cartesian product(CP) of an annulus with a small square. The spokes needs to fit tightly in holes. This is easier with round holes, so a spoke would be a spherinder, the CP of a sphere with a line.
Once this was done, a wagon is not far away. Use two of the wheel assemblies, forward and back, for a total of eight wheels on four axles. The forward assembly is on a yoke that steers independently of the body. The yoke pivots on an axle that is a spherinder so that it may move both left-right and kata-ana. That's all.