If your wheel is sound, you should be able to get it done at a tire store. Since the DS is so light, you can leave it on the trailer - just reassure them that they won't have an issue jacking it up. (Can't bring in the trailer in your case, since your tires don't inflate
But if you can get the wheels off, they should be able to put tires on them)
That's what I did - but I did call ahead first.
Other alternatives would be stores that specialize in trailers. Those are a great place to go if you have to replace a wheel (because it's rusted). Not all wheels can be replaced - for some, you simply can't find replacements.
My trailer dates from 1965, but apparently uses generic sizes so that getting wheel bearings and tires has not been a problem. Springs will be another matter - I may have to have them custom built.
Before you give up, check online as well, but be careful you understand any measurements quoted.
If you can't get a wheel or tire and really need to replace the whole setup, then one option short of getting a whole new trailer might be to replace the axle - assuming that it's bolted onto the springs with U bolts, that's relatively straight forward for a small trailer.
The biggest problems are wheels that don't have separate hubs or axles that are welded instead of bolted. But I think those are more typical for trailers for lighter dinghies.