The next step after the 4D oxcart and wagon would be a tricycle. While an oxcart is rectilinear and very much constrained in its design -- oxcarts are the same the world over -- there are a number of ways to build a tricycle. Here is a classic design.

A recumbent model.

A 4D tricycle would be a quadricycle, with the four wheels at the vertices of a tetrahedron. But to me the most interesting thing is the steering. While a quadricycle has three trailing wheels, it still has one steering wheel. While a tricycle has steering that is confined to a horizontal plane, a quadricycle would have a spherical "steering space" with two degrees of freedom instead of one. To handle this the steering shaft would be a spherinder, as it also has two degrees of freedom.

At first it seems that more handlebars would be necessary, but this isn't so. The bars are rigidly attached to the steering shaft and force can be applied in any direction. Indeed, one hand on a tiller would be enough. This continues to be true no matter the number of dimensions one may happen to find oneself.

The axle of the front wheel is a cubinder for directional stability. The front forks are a slot cut into the spherinder drive shaft to accommodate the cubinder wheel, and smaller slots for the cubinder axle.

The three rear wheels could be connected in a number of ways. My guess is a single axle that is the Cartesian product of a small circle with a smoothly curved shape. This shape would be a compromise between at one extreme a solid triangle and at the other extreme three lines from the vertices of a triangle to its center. In other words, it would be three lines that meet at a center but smoothly filled in with material so that there are no angles to focus stress. Then a triangle or two atop that upon which your buddies may stand and ride along.

The remainder -- the seat, and the load-bearing curves of the tube(s) that support it -- would be much like those seen above.