It is possible but relatively difficult. From my friends who have been in Entertainment as character performers, Disney will usually start you on fur only first. They have maybe one or two face character auditions a year that are open to CPs. (Most aren't open to them for various union reasons, IIRC.) They mostly only look for a few specific characters at a time based on need and don't always tell you what they are looking for. And if you get picked to do face, you'll still have to do fur part of the time too in almost every case - so if you want to become a face character but don't want to do fur at all please keep that in mind.
That being said, I've known a few CP face characters. One of my roommates from my first CP had a friend she worked with quite a bit who became friends with Mulan. I've also met a few CP friends of Tinkerbell as well. I've even met people who've gotten picked to be parade dancers!