Background: The COVID-19 pandemic is the biggest global health problem in the last hundred years. The efficacy of the vaccine to protect against severe disease is estimated to be 70–95% according to the studies carried out, although there are aspects of the immune response to the vaccine that remain unclear. Methods: 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 were studied. SARS-CoV-2 IgG and IgA antibodies, αβ and γδ T-cell subsets, and their differentiation stages and apoptosis were analyzed. Results: Anti-SARS-CoV-2 IgG and IgA antibodies showed a progressive increase throughout the duration of the study. This increase was the 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 a γδ T cell deficit, specifically CD8+ TEMRA and CD56+ γδ TEM, as well as lower pre-vaccine apoptosis levels. Conclusions: The results unveil the important role of γδ T cells in SARS-CoV-2-vaccine-mediated protection from the disease.