The "Dawn" is Dawn Winkler, a long-time anti-vaccination activist, who is determined to defeat AB 2109
Dawn's "83 of 105" figure was from "research" was evidently done by Amy Templeton Helstrup. Evidently she called pediatric practices in a 30-mile radius of her zip code. I'm not sure what she asked; I can imagine that you could phrase the question in such a way to get the scheduler to refuse. So pediatric practices are refusing to take new families who are vaccine-refusers.
HAPI is Health Advocacy in the Public Interest. In 2004 they sent 4 vaccine syringes for testing to Doctor's Data, a notorious lab. It's not clear to me that Doctor's Data is even competent to test vaccines.
I'm not sure what the next tweet is about. Evidently the anti-vaccine forces proposed a number of amendments to the bill, but I can't find texts for them.