The vaccination campaign got off to a slow start, from what I understand, is that despite having home grown vaccine development and being a major manufacturing centre, China had such few covid cases that they exported most of the vaccines produced to developing nations that couldn't afford the exploitative prices that western companies were strong-arming them into.

So yeah, they were off to a "slow start" by treating a world wide pandemic as if it is a world wide pandemic, and assisting countries that were left of it of luck by America.

Saying that, if cases in China started increasing they would quickly catch up. They've proven they have the infrastructure to manage it, unlike the just-in-time bare minimum western health care systems.