Well, I was partly right.
Apparently you can by SS192 (the hollowpoint 5.7x22 training round, no penetrating core), just not the SS190 armor piercing stuff.
Of course, it is a little expensive...
http://www.gunsamerica.com/guns/976328128.htm
Description: WTS new Winchester prodiction SS192 is the new designation for the training ammo for the FNP90 submachine guns and FiveseveN pistols. We were just given approval to sell this ammo to anybody, including civilians. This is hollowpoint ammo and is not armor piercing. $25 per box of 50 rounds or $460 per 1000 rounds.