I don't see any problem story- or gameplay-wise by having your party members do the rolls for you if they happen to have the best stats for it. In fact you can already do that in the game, it is just tedious. As it is right now, you can close the dialogue with one party member, switch to another one and start the conversation again.
Just look at the Pathfinder game, or Neverwinter Nights 2 as HunterRayder93 already mentioned. Nothing story- or gameplay-breaking there. You don't even need to add fancy animations or additional voice over lines to the game since all your answers aren't even voiced to begin with.
It is just that simple: You see a skill check, you hover your cursor over it, the game tells you "Lae'zel: Intimidation +3", done.