Two thoughts come to mind. I built a block cart for my kid's blocks
many years ago (cart is still being used by grandchildren). It had a
1/2" plywood base. I simply got some industrial grade casters and
screwed them on to the bottom. Fairly large wheels -- maybe 1 1/2"
diameter. You might want to go larger in your case.

Also, somewhat later, we made a "go cart" of wood. For wheels, I used
lawnmower wheels. However, I used large (1/2") long machine bolts. To
secure them to the base, I used a pair of "U" bolts to clamp the bolts
to the 2x4s that were the axles. I cut a small groove in the 2x4 to
reduce "wiggle," and the "U" bolts retained the mounting bolts in the
groove. I used some 1/2" washers to space out the wheel from the end
of the 2x4. The "go cart" generated a sufficient number of scrapes
and bruises....