There is no proof, you need to read the whole document cited by this Steve Kirch.
Systemic and mucosal IgA responses are variably induced in response to SARS-CoV-2 mRNA vaccination and are associated with protection against subsequent infection | Mucosal Immunology
"Systemic and mucosal IgA responses are variably induced in response to SARS-CoV-2 mRNA vaccination and are associated with protection against subsequent infection" Title page
"These data suggest that COVID-19 vaccines that elicit a durable IgA response may have utility in preventing infection."
and
"Collectively, our data show that a SIgA response is induced in ~30% of participants who received 2 doses of a SARS-CoV-2 mRNA vaccine, and that IgA may play an important role in protection against infection."
"Therefore, a local SIgA response to Spike/RBD is produced in response to mRNA vaccination in some participants."
"In summary, we provide evidence that anti-Spike SIgA are induced and maintained in the saliva of approximately 30% of mRNA vaccinated participants and that high anti-Spike/RBD serum IgA levels are associated with protection against subsequent breakthrough infection."
Bottom line is the vaccine does give protection against covid despite Kirsch's twisting the evidence.
In the second paper he sighted
Serum and Salivary IgG and IgA Response After COVID-19 Messenger RNA Vaccination | Vaccination | JAMA Network Open | JAMA Network
"The findings of this cohort study suggest that mRNA vaccination was associated with mucosal immunity in individuals with prior SARS-CoV-2 infection. Further studies are needed to determine the association between SIgA levels and prevention of infection or transmission."