Wing Chun is uncommon in MMA for three major reasons. First, its practitioners generally shun sparring. Second, its practitioners generally shun realistic wrestling, throwing, and groundwork practice. Third, the skill Wing Chun focuses the most time on – hand trapping – is a small part of fighting in general, and is easily overshadowed by boxing or clinch work (that is, wrestling in any of its many forms) unless the WC stylist has extensive experience applying WC to rulesets more open than sticking hands.
The results of trying to apply just Wing Chun sticking-hands experience to MMA are consistently poor: they get taken down and smashed or choked. To be effective in MMA one needs to train effectively in all the aspects of fighting, not just the hand-trapping range so popular in Wing Chun. Wing Chun's singular strategy of closing the distance while neglecting takedowns is simply not going to work, whether in the cage or in a street fight, against an aggressive, athletic opponent who closes the distance even further, to a clinch.
Shawn Obasi explicitly represented Wing Chun in MMA. He did OK on the amateur circuit, but his pro debut was not so effective. Watching his amateur fights, I do not see Wing Chun in action. For example, in this fight, he does very well against what appears to be a grappling specialist, but that seems due to his sprawl, athleticism, and guard (from some BJJ he's done). His striking seems, to me, singularly unimpressive (though probably better than mine!).