There are dozens of sources for new tires. Carter Products, Iturra Design and Delta (Rockwell) are just a few.
Running without tires probably wouldn't hurt the blades, but you'd sure tear up the wheels. Also, wheels that take tires are designed to have the tires crowned - which is what keeps the blade from running off.
That was just it. I looked at about a half dozen of the first hits I got with a google search and could not find exactly what I wanted. This a rather old Rockwell and I didn't find Delta's site to be very helpful.
Yeah the wheels are already showing wear. The saw is offline for now. It also needs a new blade of which I have several in stock.
Looking at the profile of the old tires it does appear that it was crowned.
I have never welded a bandsaw blade before (I have the manual). Would you recommend welding or silver brazing?
There are three instances in which you'll want to join a bandsaw blade:
when making up a new blade from roll stock
when repairing a broken blade
when cutting and joining a blade to cut an inside shape
For #1, get your blades welded to length by the manufacturer or distributor. They will do a much better job than you will.
For #2, this doesn't happen all that often. Usually, by the time a blade breaks it's pretty well worn. But you can repair by either welding or silver soldering/brazing. Welding is probably stronger, and doesn't require overlap scarfing, but it is usually done with specialized bandsaw welding equipment - not your basic home welding setup. For occasional breaks, I'd save up a few blades and then have them re-welded at a bandsaw shop.
#3 is the real reason to want this capability yourself. If this is something you don't need, skip it. Most users don't need it. If you do, though, you should consider a dedicated blade welder for heavy use, and silver soldering/brazing only for occasional use.