Thank you for the replies. The movement is up-down or front-back, not in-out. Both sides move exactly by the same amount in the same radial direction. No grinding or rough rolling when I spin the wheels. If my car wheel moved to the same degree, I would replace the hub or replace the wheel.
By looking at the construction, I can tell tow dolly wheels are not designed like a car. The entire "axle" is a square bar inserted into a square pipe with plastic shims between them and 2 bolts holding the pipe to the bar. This is an older Demco model with pivoting wheel pan and hot dip galvanized. I bought this used and don't know the mileage. Previous owner knows no history, either.
I have a related question. The replacement wheel hub (Demco brand) is very expensive. For about the same price, I can add an electrical brake using non-demco parts. But how do I go about it? Once the sealed hub is removed, I will have 4 bolt holes, 2 3/8" x 3 1/8" pattern. Also there is no spindle. What can hold a hub if no spindle?