Discourse, the software, does support custom emoji, and in fact someone even ported a bunch of the vBulletin ones over. On a “normal” Discourse forum where the forum admins/owners are available and interested, it wouldn’t be very hard to set up.
However, I think in our case… um… the SDMB is apparently just hiding in some forgotten corner of our owner’s accounting, mostly ignored, and from what I understand, it’s best not to poke the beast…? See the discussion at Can we consider allowing direct image uploads?. I think the overall consensus in that thread is (my paraphrase) “We’re lucky to have the forum at all, hosted at no charge to us, and it’s best if we don’t do anything to risk that”.
On the other hand, hopefully the theme stuff is something our moderators (as opposed to the site owners) can (and apparently did?) fix on their own.