I think there are regional differences. An older thread debated if western riders would post, and the responses seemed to vary with the part of the country.
How well it works depends in part on your choice of saddle. Old style western saddles were designed to put the foot well in front of the hip. Not exactly in a chair seat, since the leg was long, but still well forward.
If you ride with a long leg and your feet forward, posting is darn hard. At least, visible posting is. I would argue you can post because posting only requires that weight be out of the saddle, not that it involve a jack-in-the-box spring. But if you are riding in a saddle & style like the guy below, you aren't going to have much visible motion...other than the ends of your handlebar mustache! John Jackson, Matador range boss, looking out over a herd of 5,000 cattle at a roundup. Matador Ranch, Texas.
, 1906 Erwin E. Smith Collection Guide | Collection Guide
At some point - maybe the 60s or 70s - folks began riding western saddle with their heels more underneath them. When that happened, posting began to make more sense. It is tough to post when your legs are almost straight, and with your feet out in front. And western saddle makers began to make saddle to fit that style of equitation. So now a western saddle may or may not put you in a 'chair seat' - or 'Harley seat'.
If it does, and you insist on posting, you may end up looking like my youngest:
Dad gets pretty grumpy when he watches this, but she has the vast knowledge and confidence that goes with being 14...
So yes, some western riders post. Others do not. The saddle may or may not make it hard to post. But if you're writing a historical novel, and you say John Jackson of the Matador Ranch was posting his horse...well, it will be like a novel set in the 60s where the hero pulls his cell phone out of his pocket and dials 911...