I’ve looked online for this service so I can get an older revolver converted, but only a couple results show up, none of them being around here. Are there any shops near Seattle, or even just the whole state, that will mill a revolver for clips?

Good grief all you need is a guy with a lathe. I would think that there is someone here that could do it for you BUT I am remembering that some revolver makers put ancillary serial numbers on the flats between the cylinder bores.

What revolver?
I've done 2 guns, 44 629 came out great but used a M29 cylinder because I didn't want to screw up the SS cylinder. Also have to turn down the ejector star. Used Performance center clips. My dad did another model 29 and used another brand of clips that were a little thinner. Did a 38/357 (Rossi Cyclops) also but the area between the bores was so small that the stainless steel moon clips (SS only ones I could find) deformed enough when loading to make it very difficult to get last round in the clip. Had to make sure and use brass with deep grooves. Eventually put a regular cylinder back in. Larger frame 38 would be fine.

I'm not sure how you'd get it all done on a lathe, as the inside relief for the moon clip comes in further than the extractor star. I've only seen it done on milling machine.
On my guns, the numbers marked on the cylinder don't correspond to a serial number, but are just to know what hole you're shooting out of every time. I've never bothered to see if some holes shoot more accurately than others, but I guess if you were really detailed on those issues, it'd make the job easier to track.

https://www.youtube.com/watch?v=WVfHKkAAE50

I would strongly recommend just sending it to TK Custom. Really fast turnaround, all you need to send is the cylinder and crane assembly. Friend of mine got his 629 done for competition and he stomps with it.

We've done 3 on lathes. Just go slow and easy. But yes, if you don't have a lathe, send it to TK