I have a Home Depot type of dolly that has a flat tire. Tried pumping air
into the valve... couldn't get any pressure in the tire. Doesn't seem to
have any nails in it. Couldn't see any tube inside either. Reminds me of a
regular car tire.
I know it sounds silly, but I thought of bringing it down to a tire shop and
having them look at it. Or, would I be better off taking off the wheel and
buying a new wheel/tire unit?

The tire thickness and wheel build is similar to that of a regular car tire.
Get the feeling I would need a special machine to take it off and put it
back on again.