They just signed Vazquez to an extension a month ago, locking him up through 2021. I think they have until Pedroia gets back to decide between Swihart and Leon. And I don’t see them putting any resources into adding anyone else.
It seems likely that they stick with what they have, for lack of alternatives if nothing else. Vazquez's extension means we want him on the roster for an additional three seasons after this one, not necessarily that the FO wants him starting 110 games in each of those seasons. He's not cost prohibitive to keep as an active backup if we are able to find a catcher of the medium term somewhere. At the same time, with the top 6 this lineup runs out there, I'd be surprised if improving C production (and JBJ's bat, for that matter) is a high priority.
If they do look to the trade market anyway, two guys who might become available are Devin Mesoraco (CIN) and Welington Castillo (CHW).
Mesoraco had big age 22 (2010) and 23 (2011) seasons to vault up through the minors and eventually exploded for a 4.5 WAR season in 2014. He missed almost all of 2015, 2016, and most of 2017 with injuries and has been passed by Tucker Barnhart as the everyday guy, who like Vazquez also got locked up this offseason. Mesoraco, meanwhile, is a free agent at year's end, but signed this year for $13m. Not too difficult to see him get moved by a team that's already done for 2018. Would I be terribly excited about him? No, and it might be Dan Butler time quickly given his DL history.
Castillo is less likely to be moved since he was just signed to a two year contract and the White Sox don't really have a top prospect they ought to be giving playing time to instead. Although he has been over 100 wRC+ just twice in the last five seasons (109 in 2013 and 113 in 2017), he's been a 1+ WAR player in each of his last six seasons (2+ WAR in three of those), and he's much less of a durability risk than Mesoraco. Not a good reputation defensively, but you can see where he is a better fit as a 60/40 type with Vazquez starting against all lefties and select righties. This seems like a guy we would have given strong thought to signing in the offseason if we hadn't committed to giving Leon another shot. 7.25m in 2018 and 2019 followed by 500k buyout or $8m in 2020.
For LT catchers, I think you have to look at St. Louis having both Carson Kelly and Andrew Knizner stuck behind a not-falling-off-the-cliff Yadier Molina. But it's extremely difficult for two contenders to trade in season, neither one addresses 2018 production, and what would we even have on hand to send back.