How do in-race repairs work in Lemons with regard to the $500 limit?

Pretty much open.  I believe there's actually a section of the rules that mentions entire spare parts cars.  By the time you have a failure and swap the parts, you're pretty much screwed for finishing position anyway.

There's a lot of grey area.

Show up to your first race with brand new radiator hoses that don't appear in your budget, you'll get nailed. Replace yer old busted hoses with brand new ones when they break during that race, and remind the judges at the next race that you had to replace those hoses when they broke AT the last race, and they're not going to give you any grief. There's a limit to how far you can take that, though.

If you have a spare freshly built and cheaty engine mated to a freshly rebuilt transmission with a new race clutch and a lightweight flywheel and you conveniently swap it into the car when your engine spins a bearing in the 3rd lap... If you're quick with the swap, you could very well WIN that race (doubtful, but possible if it's a fast car with good drivers and pit strategy)... but it's probably not going to fly at your next BS inspection.

Bring all the spares you can think of and have room for. Once the green flag drops, keeping the car on track is the name of the game no matter what you have to put on the car to do it. Just be aware that you might have to account for parts, especially new or cheaty ones, at your next BS inspection.

Generally, if you're replacing old junkard parts with other old junkyard parts, you'll have nothing to worry about.

Our 3rd race out we had to replaced the rear-end after snapping the watts-link during a previous race. We didn't present a budget or docs for the car as we didn't feel the need nor did we really even bother with it due to it being virtually impossible to track down a replacement that matched.

When we did the BS inspection they had asked us the routine of questions they generally had. "Anything done to the car since the last race, Any mods/upgrades... etc etc). We gladdly admitted to replacing the rear end and putting a new trailing arm on and they replied with, "We don't care about the stock/ OEM crap, get out of here." ... "oh, by the way, what's up with the shiny altenator." knowing it was just a joke to mess with us, we just gave some lame-ass excuse. Even though that's how it looked when we got the car originally.

There's a lot of ways it could go so just take what i mentioned about our team's previous racer with a grain of salt. For a lot of oem type stuff they may just let you slide but if for instance you have like an Olds Intrigue (probably a Grand Prix more likely) and have upgraded to the a dual piston front caliper, say from a Saturn Vue, probably not so likely to get off the hook on that one. I know brakes don't count toward it but I'm just using it for the sake of an example.