This claim originated from a hearing of the Texas State Senate Committee on State Affairs.
As a witness, Dr. Angelina Farella, a pediatrician, made this false claim along with several others, including "we have in excess of 4000 deaths and this thing has not been pulled yet." There is no evidence that the vaccine has caused "4000 deaths".
There is no evidence to support this claim.
Preclinical work on the Pfizer BioNtech mRNA vaccine in mice and macaques is described in this paper:
"The responses elicited by the vaccine candidates reflect a pattern that is favourable for vaccine safety and efficacy, which provides added reassurance for clinical translation"
Fact Check-COVID-19 vaccines did not skip animal trials because of animal deaths (2021-06-01)
Covid-19 vaccines were subject to animal trials
Instagram Posts Spread Texas Lawmaker’s False Claims on Vaccine Testing (2021-05-19)
"Had the animals died during this process... the human trials would have been immediately halted."