What else will need replaced when I replace my wheel set on my vintage KHS road bike? I am needing to get a new wheelset for my vintage KHS road bike. I have been looking into it and to get my 6-speed cassette to work with the Shimano 501 wheels it looks like I will need to get a spacer. Is this true or do I have to upgrade the cassette as well? If the cassette gets upgraded, what other parts would need replaced to get the bike running again?

Thanks in advance!
You don't say what components it has now (Shimano or Suntour and which specific parts, right shifter indexed or not, etc), what the rear end spacing is (probably 126 if it's a 6spd cassette), and whether you care about indexing if you do have an indexed right shifter.

But, if you have a 6-speed Shimano Uniglide cassette, all but the smallest cog can in fact go on a modern Shimano-compatible wheel. More info on this here and here. You'd then get a modern smallest position cog and a lockring, and spacers for behind the cassette to take up all the extra room. You'll also need some very thin spacers to get the cog-to-cog spacing right on the small cog. Again according to Sheldon's numbers, you need .5mm worth there if you use a cog from a 7spd Cassette, since that cog will have a built in spacer for 5mm cog-to-cog and you want 5.5. Shops do have cassette spacers that thin, but they're very seldom used now so you may have to hunt.

To be clear, the main point of doing this would be an interesting and esoteric bike geek project. After you go through all the trouble of setting it up, you'll have to get another vintage cassette when you finish wearing out this one. (Although it may be possible to make a new one by re-using the spacers from your old one on a new 7spd cassette, or machining your own spacers, etc... again, esoteric.)

The simpler thing by far in a lot of cases is just going to be use a new 7spd cassette and 7/8-speed chain, which will still need a 4.5mm spacer, (plus a 1.8mm spacer on an 11spd freehub, if you're looking at other wheels too). That plan may or may not work seamlessly depending on what else is on the bike, but it probably will, and for cheap. You'll be running it friction unless you add a 7spd indexed shifter, but those are available both new or used/vintage for not much either.

If this is a 126mm spaced steel bike then it's more or less okay to just jam a modern 130 wheel in there and go. If you want to take the best care of the frame possible, you should respace it though. If it's 120 you should definitely respace it, and if it's not steel you shouldn't do it.