Part of the problem is China's vaccine is substantially less effective than the mRNA ones. "Just make vaccines available" works less for them than it does in, say, the US.
https://www.globaltimes.cn/page/202103/1219245.shtml
I was in Taiwan for the bulk of Covid and the only possible source of a Pfizer vaccine was from China’ Fosun group. It looks like Fosun had millions of doses sitting around for no one in mainland China to take
https://www.bloomberg.com/news/articles/2021-12-12/billionai...