I think から
here is "a particle which indicates a starting point or a source" (A Dictionary of Basic Japanese Grammar, p.176). 顔面から
is describing a verb of motion (コケる, to fall).

(My apologies for the bad drawing and handwriting! It's supposed to say [顔面]{がんめん}から・[先]{さき}に・[地面]{じめん}に like in @Chocolate's comment.)
So the motion described by コケる
is from the face, toward the ground. 先{さき}に and 地面{じめん}に aren't part of your original sentence, but I think you can use them to describe the motion.
I don't think you can use に
here. I think you're trying to understand by substituting English words for Japanese ones. If you do that, your sentence sounds like "I fell from my face", and you're wondering if you can substitute に
because you think the English phrase "I fell on my face" sounds more natural. Unfortunately, this doesn't work, because although コケる
expresses falling, it doesn't do it in quite the same way as the English word "fall".