##On Mechanics
If, when you ask other players to participate, you're requiring rolls against social skills, then they have a valid complaint: they should leave the talking to the Face, and stay out of it.
If you really want to involve other characters in the role-playing scenes, don't make them roll against the skills they're bad at. Instead, you should grant advantage to the Face's rolls due to the assistance and participation of the not-particularly-social characters. This way, the non-social characters' poor skills aren't dragging down the characters' ability to get things done. Also, make sure the players know how you're going to handle this. If you decide to add this "assistance = advantage" mechanic, but don't tell the players, they might still think that you're working against them.
Finally, roll less. The majority of conversations require no roll at all. For those where rolling is required, just one or two rolls should cover the entire conversation, taken at the point where the NPC is making a decision.
##On Socializing
Talk to the players about this. Tell them that you'd like them to participate in the role-playing scenes, and why, and how you're going to handle the mechanics.
Work with each player to make sure that their characters are fleshed-out, with a background and personality. Make sure that both the players and their characters care about your plots.
As the players start to role-play, don't punish them if it turns out badly the first few times. They'll just learn that role-playing is the Face's job and back out again. By punishing them, I mean let them succeed at what they're trying to do – a blundered interaction may look like an "interesting development" to you as the GM, but to them it will just look like everything went wrong and they shouldn't try this again.