I'm in a similar situation, but unfortunately, it is not possible to visit relatives in China at the moment absent compelling reasons (e.g. very ill close relatives).
Are there any additional restrictions for traveling to China with a valid visa?
Yes. A valid visa is only an entry authorization,
vaccination at the moment does not allow exemption from other applicable procedures.
You still need to have negative test results and green health code. And you still need follow local public health protocols which involve mandatory testing and quarantine measures that can be very long, expensive and somewhat invasive. At least a week of quarantine at designated hotel at own expenses is required (the minimum policy for people first entering Shanghai with Shanghai addresses, otherwise it's two weeks), depending on your destination you may also face up to four weeks of follow-up supervised quarantine.
Meanwhile, the airline boarding requirements by China for Certificates of Negative Nucleic Acid and Anti-Body Blood Tests Results remain unchanged. After entering China, please abide by China’s regulations on quarantine and observation.
http://www.china-embassy.org/eng/visas/zyxx/t1861379.htm
Is it possible to go with an non-Chinese vaccine (Pfizer, Moderna, etc.) as I don't think it is possible to get the Chinese vaccine in the US.
No.
The "relaxation" measure is mostly a political statement with respect to Chinese-produced vaccines. It also facilitates return to China of foreign citizens already in China (who can get Chinese vaccines) who visit other countries temporarily.
It does not apply to Western vaccines and it is not likely that Western vaccines would be accepted before some sort of reciprocal recognition of Chinese vaccines.
Also, I need a visa for my 2-year old daughter. Are toddlers also required to take the vaccine?
"Yes", but toddlers cannot legally get Chinese vaccines anywhere as they are not approved for that age category. Although "urgent humanitarian reasons" might be enough to exempt her from the requirement if you are vaccinated, that's for the consulate to decide.
For English summaries of Chinese policies, I find https://www.china-briefing.com/news/how-can-foreigners-enter-china-under-the-covid-19-travel-restrictions/ to be fairly comprehensive. But it is still better to read original Chinese policies on consulates' websites or call them (if your local consulate is nice enough).
