Some races - I think female elezen and female midlanders - will sit sidesaddle on all of the primal-ponies, aswell as the unicorn.

For other races I could see the following work well:

Archon Throne (flying chair that you can buy for 750k MGP)
The two Broken Heart chairs (from this years valentines event, they'll be back in the mogstation next feburary, most likely)
Flying Chair (Crafted, using some seeds you need to crossbreed)
Mikoshi (from the Namazu Beast Tribe once you completed all quests and have some Koban left)
all the "System"-Mounts that you get from various PvP-achievments (those glowing CPU-balls)
The Ahriman and the Disembodied Head (needs PotD)
Bomb Palanquin (from the Cobold Beast Tribe)
Adamantoise (200k MGP)

...since the sitting animation varies from race to race and between genders, some of those might work better than others for each character. I'd expect all the flying chairs to work well enough for each race/gender, aswell as the Mikoshi (since you're standing on that). I know that the Bomb Palanquin can be weird for certain race/gender combinations (it brings back painful flashbacks of being a male elezen - they sport a fantastic example of manspreading on that thing - looks horrible, but looks especially horrible with coats/skirts), but afaik female Mi'qote got a decent sitting position for it!