Well... you can paint chrome if preped properly and with the right primer, but you wouldn't want to b/c if the chrome underneath begins to flake, crack etc. it would destroy your paint job. I suggest you call around to your local chrome shops, and they can strip the chrome for short money. They will dip it in an acid bath that will only eat away the plating. Then I would paint it the day you get it back, you would be surprise how quickly rust sets in on a raw piece.
I should have read better, I didn't realize it was the ABS plastic grill chrome you were talking about. I don't believe it is actually chrome on that just a chrome like finish, but it does flake and peel so it's not a good idea to paint. What I did to get a black grille is I bought an aftermarket bare plastic front grille, the kind found on some sierras. Preped it with sanding, Plastic prep, & proper Primer, then Painted it. If I remember right the grille didn't even cost me $100, and fit perfect.