Yes, going up to a 12" wheel is a common upgrade and a good idea. You will reduce wear on the tires and on the bearings and increase the load capacity of the wheels. 4.80 x 12" would be the next size up. You could also go to a 5.30 x 12" which would be a wider tire, but probably a bit overkill. Make sure you get a wheel that has the same bolt pattern as your hubs since they could either be 4 lug or 5 lug. As far as the fenders are concerned, there is probably enough clearance that you won't have to make any modifications, but even if they end up close, it should be pretty easy to relocate them. At worst, it would likely just be a matter of drilling a couple new holes or modifying the fender bracket.
If you're unsure the last time your trailer has been serviced, it would probably be a good idea to pull the hubs and inspect the condition of the bearings & grease. Also check the axle and U-bolts for cracks or excess rust. I can speak from personal experience here that trailer breakdowns are not fun. Also make sure you carry the proper size lug wrench for your trailer lugs - this may not be the same size as your car's lug wrench.