In teaching a course about coffee houses, we went to lots of independent coffee houses, and a few chains. The independents were overwhelmingly – no, uniformly – left of center in their feel. The corporate ones were more politically neutral, but even Starbucks feels more Democratic than Republican. I have taken to asking people if they have ever been to a Republican coffee house. So far, no examples. It seems possible that there are some evangelical Christian coffee house ministries that might serve GOP lattes, but I haven't met one yet.
Indeed, it is hard to imagine what, exactly, a Republican "third place" would be like. I know about some redneck bars that have a right-wing stance, but I don't think they are really for party regulars and primary voters. And I suppose the average country club restaurant is more R than D. Still, it is hard to think of the kind of place where Republicans would normally go to hang out and talk about the world with strangers who gradually become acquaintances.