Vaccinations are available nationwide inside of China. To the best of my knowledge, foreigners need an appointment and locals will need at appointment at some vaccination sites, but not others. Right now, I don't see any big pushed to "Get vaccinated today!", must likely because it will take awhile longer to produce enough vaccine to get everyone one done. Instead, promotions are small and local so that supplies can keep up with demand.
The latest data I can find shows 872 million doses administered, but doesn't break that down to how many people are partially vs fully vaccinated. Assuming a target of 1.2 billion fully vaccinated (which would be a solid percent of the public) and 2 doses, we'll need to get 2.4 million injections done, so we're very roughly about 36% of the way there. Considering the number went up over 500 million in the last month, even if things level off, it won't take too many more months to get almost everyone fully vaccinated.
My best guess is that not all teachers will want to come back, so I expect you could probably find some openings.