Can you run out of Copper? (the currency)

[Supa]

You only seem to gain it for the first time you beat X medals in the track. And every offical will cost you 10.

Or can you grind it online or smthing?

[Chic-Fil-Bae]

If you bought TrackMania United, You get 50 coppers per day and more if haven't played in like a week or month for a present of returning. That's all I know.

[Arkive]

You get 50 coppers daily (more if you register TrackMania Original or Sunrise keys) and you can get more through a few different ways. There's the ManiaLoto (go to your bookmarks menu), which has some basic gambling minigames; I've won some from it. There's beating medals in an official run (which you mentioned already). There's also having coppers given to you; this could be from servers giving out prize money, winnings from competitions, or somebody giving them to you (for whatever reason). I'm pretty sure that's all you can do to get them (there's also charging for your creations, but I don't know if that's a reliable source of income anymore).