Most of the horses that I ride are smooth enough that I can sit the trot but my Percheron travels so animated that sitting the trot is impossible and I have to either post or stay standing in the stirrups. Most of the cowboys that I know will post when they are in a long trot because it is just easier. No, I don't think that western shows allow posting but there may be some classes that do.
I ride both English and Western. It's true that posting will get you disqualified in western shows, but many western trail riders post on trail. It's much easier on the horse's back over long distances. My last riding instructor (who is the head western riding instructor for the USA's top Equine Science University) told us to post when we are warming up horses at a working jog. This is so much easier on their backs. I also worked at a horse camp for the past two years and we taught the kids to post in a western saddle because it's just more comfortable for some people, especially on trail. With some of the horses and ponies I rode there it was nearly impossible to sit the trot on them, WAY too bouncy. And some horses would not trot until I started posting. That's just the temperamental nature of camp horses :\.
So no worries, lots of people do it. I think more people should. If people are giving you funny looks, pay them no mind. If you and your horse are comfortable that's all that matters.
I don't ride western, but I do know that my instructors tell me to always warm the horse up in rising trot before doing a sitting trot, because it requires more strength through their backs. If you're rising properly, there's less weight on their backs (and the whole idea of rising on the correct diagonal is to allow their inside hind to drive forwards more easily) and they can move a little more freely.