DNA and RNA vaccines (nucleic acid-based vaccines) are a promising platform for vaccine development. The first mRNA vaccines (Moderna and Pfizer/BioNTech) were approved in 2020, and a DNA vaccine (Zydus Cadila, India), in 2021. They display unique benefits in the current COVID-19 pandemic. Nucleic acid-based vaccines have a number of advantages, such as safety, efficacy, and low cost. They are potentially faster to develop, cheaper to produce, and easier to store and transport. A crucial step in the technology of DNA or RNA vaccines is choosing an efficient delivery method. Nucleic acid delivery using liposomes is the most popular approach today, but this method has certain disadvantages. Therefore, studies are actively underway to develop various alternative delivery methods, among which synthetic cationic polymers such as dendrimers are very attractive. Dendrimers are three-dimensional nanostructures with a high degree of molecular homogeneity, adjustable size, multivalence, high surface functionality, and high aqueous solubility. The biosafety of some dendrimers has been evaluated in several clinical trials presented in this review. Due to these important and attractive properties, dendrimers are already being used to deliver a number of drugs and are being explored as promising carriers for nucleic acid-based vaccines. This review summarizes the literature data on the development of dendrimer-based delivery systems for DNA and mRNA vaccines.