The wheels right now aren't welded on, they seem bolted. What you'd need is tools to unbolt them, and you'd need some rubber wheels to bolt on.
Welds are only as strong as the quality of the welder and the material; the types of metal need to be the same and the base plate for the wheels has to be thick enough to withstand the force of movement. This makes it difficult to advise you from far away - and I'm no metal alchemist so I really can't tell from looking at it.
With bolts, they're easier to replace if something breaks. You don't want to haul this thing on three or two wheels to your car, to get it to the welder, have him grind the old ones off, to install new wheels. Those bolts would work just fine.
To do a proper weld, paint has to be removed entirely, and you'll have to repaint it yourself (or have it done) as well. Something to consider, because these things are usually magnetically painted (or whatever it's called) with heavy duty paint. To keep the quality, you'll have to think about a 300 USD paint job on top of the welding process.