As others have already pointed out, this is false. Vaccines do not cause autism. The CDC has a great information that is summarized here, but they also link to lots of sources in that post.
On the folks involved in the X post you've linked: I think that the subject of the attached video, Jamie Melillo Juarez, is interesting. Searching up her name shows an amazon listing for a book authored by her titled "Hope for autism". The description of the book says: "A diagnosis of autism is the single most heart wrenching experience a family can go through", which feels really gross to me. Autistic people are just people, a child isnt dead to the world following an autism diagnosis. I think most autistic people would be upset if you asked them how their family was coping with their autism diagnosis. I can't access much more information on Juarez as far as here counseling practice, but the description of the book she is selling leads me to believe Juarez is taking advantage of parents who are struggling to adapt to a child's autism diagnosis to sell her book or advertise her practice.
On the topic of anti-vaccine voices, I did a little digging into the history of the claim that vaccines cause health complications. I've linked below some information on Andrew Wakefield, the first (formerly) credited Doctor I could find making these claims. There's information about his published research and subsequent falling out, and information which may provide insight to his motives. I have also linked to an article from the College of Physicians of Philadelphia which traces anti-vaccination movements all the way back to the first vaccines in the 1800's, which I found very interesting. Not necessarily related to the fact check, but interesting all the same.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3136032/
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6181752/
https://historyofvaccines.org/vaccines-101/misconceptions-about-vaccines/history-anti-vaccination-movements