Do insurance companies file civil suits to recoup payouts?

My car got keyed at work. I know who did it, but criminally it doesn't seem as though there's enough direct proof to prosecute. There is, however, very very likely enough to file and win a civil suit. The trouble, obviously, would be actually getting the money.

If I claim the damage through insurance (state farm), would they file a civil suit against the individual to recoup their payout?