Background: The effectiveness of the anti-COVID-19 vaccine remains a matter to be clarified. Methods: The humoral and cellular immunity after the administration of three doses of the Pfizer–BioNTech and Oxford AstraZeneca vaccines against SARS-CoV-2 over one year and the appearance of post-vaccination COVID-19 have been studied. Results: Anti-SARS-CoV-2 IgG and IgA antibodies showed a progressive increase throughout the vaccination period. This increase was greatest after the third dose. The highest levels were observed in subjects who had anti-SARS-CoV-2 antibodies prior to vaccination. There was an increase in CD4+ αβ, CD8+ γδ and TEM CD8+ γδ T cells, and a decrease in apoptosis in CD4+CD8+ and CD56+ αβ and γδ T cells. Post-vaccination SARS-CoV-2 infection was greater than 60%. The symptoms of COVID-19 were very mild and were related to γδ T cell deficit, specifically CD8+ TEMRA and CD56+ γδ TEM, as well as lower pre-vaccine apoptosis levels. Conclusions: Those results demonstrate the important role that γδ T cells play in SARS-CoV-2 vaccine immunization.