You shouldn't use self-tanners for the body on your face because they often contain jojoba oil, which is super moisturizing, or coconut oil and cocoa butter, which are incredibly rich in fatty acids. If your face is oily, avoid oil-based facial tanners.
Instead, look for a mousse or foam-based formula, not a cream; they're lighter and dry faster. In addition to foams, the options for acne-prone skin include towelette-type pads for the face, drops you can add to your moisturizer, clear face mists that bronze, drops that fade by the end of the day that you'll need to wash off, and bronzing face serums that moisturize without oil.
As you're looking, do a search for "non-comedogenic" products because they're formulated to not clog pores, even though they may contain small amounts of oil. For example, some non-comedogenic facial tanners have cocoa seed oil or raspberry seed oil but you can still use them with confidence. In your research, also pay attention to product suggestions — does it work better with warm skin tones, can you spray it on over your makeup, or do you need to use it before sleep for best results?