I ran into this when I applied for away rotations, too. Two of the places I was interested in required a background check of some sort. One just wanted an official form from my local sheriff's office, confirming that there was nothing on my record. The other one wanted me to get a very extensive background check (7 years+, covering everywhere I've lived) through the FBI. I almost threw in the towel with the latter (especially when even the FBI didn't know what I was talking about... "background check" means nothing to them -- you have to say "identification record check"), but I went ahead and jumped through their hoops, and now I'm really glad I did; I was accepted as a visiting student, and I'm really looking forward to it.
I was told that it's only a matter of time before all medical schools start doing background checks on their own students when they start as first years; that will eliminate this mess for future away rotation hopefuls. Not that that helps us this year... 🙄
If your school does background checks already, then the place you're applying to may just need a letter from your school. Otherwise, call the place you're applying to and find out if they just want something local and recent or if they want a major FBI thing or something in between (a school that I ended up not applying to wanted a letter from my state police).
To answer your question about time... it took about 10 minutes to get a form from the sheriff's office, and it took about 6 weeks to get a letter from the FBI. If you need something from the FBI, you'll have to go to your local police station or whatever and get your fingerprints done -- takes only a few minutes, but it's yet another errand you have to try to squeeze in during business hours.
I hope this helps! Good luck! 🙂