Every largish city has somewhere that arcade machines get repaired. Where I live there (~350,000 population) there used to be a few but now there is just one dude that runs a repair/distribution company for arcade machines. He does a lot of private trade these days.
If you go along to any repair shop, if it's anything like my local one, there will be a room with a thousands of mainboards for a variety of arcade machines.
As time has passed a lot of games have gone in and out of favour. By removing their mainboards and replacing with a newer mainboard from a more popular game one could save money by keeping the original cabinet. A lot of mainboards were stored in back rooms, cupboards, or just piled up in the corner.
If you feel so guilty that you couldn't build a MAME cabinet without legally owning the roms I would tell you to go and buy the mainboard. You will find they run at no more than $5 each - at least for the games made in the 1980's and early 1990. As you move into the late 1990's you will start coming across games that also require a hard drive for storing game assets (tilesets, audio and video samples etc). It might be a little harder to get all the physical assets to justify to your needs (I think Killer Instinct used a hard drive).