Sometimes the dice get ... playful. Edge can sure help you pull through when that happens. That said, missions generally not pushed too hard (given that they never know what the group will be like), so it may be possible. I did go through online play of a couple of missions with a character with the Bad Luck negative quality, where using edge would be a last resort, and got through the two missions without using edge. Hardly optimal, but you don't really need to be optimal for missions IMO.
Mind you, how would you manage zero edge? You start off with at least one point.