Severe acute respiratory syndrome coronavirus 2 virus (SARS-CoV-2) infection, which causes coronavirus disease 2019 (COVID-19), has led to many cases and deaths worldwide. Therefore, a number of vaccine candidates have been developed to control the COVID-19 pandemic. Of these, to date, 21 vaccines have received emergency approval for human use in at least one country. However, the recent global emergence of SARS-CoV-2 variants has compromised the efficacy of the currently available vaccines. To protect against these variants, the use of vaccines that modulate T cell-mediated immune responses or innate immune cell memory function, termed trained immunity, is needed. The major advantage of a vaccine that uses bacteria or viral systems for the delivery of COVID-19 antigens is the ability to induce both T cell-mediated and humoral immune responses. In addition, such vaccine systems can also exert off-target effects via the vector itself, mediated partly through trained immunity; compared to other vaccine platforms, suggesting that this approach can provide better protection against even vaccine escape variants. This review presents the current status of the development of COVID-19 vaccines based on recombinant viral and bacterial delivery systems. We also discuss the current status of the use of licensed live vaccines for other infections, including BCG, oral polio and MMR vaccines, to prevent COVID-19 infections.