Young adults had been widely perceived as a low-risk group for COVID-19 severity; therefore, they were deprioritised within the mass vaccination strategies as their prognosis of COVID-19 infection is relatively more favourable than older age groups. On the other hand, vaccination of this demographic group is indispensable to achieve herd immunity. A cross-sectional survey-based study was used to evaluate the side effects of mRNA-based COVID-19 vaccines among university students in the Czech Republic. The validated questionnaire was delivered in a digital form, and it consisted of demographic data; COVID-19 vaccine-related anamnesis; and local, systemic, orofacial, and skin-related side effects’ prevalence, onset, and duration. Out of the 539 included participants, 70.1% were females and 45.8% were <23 years old. The vast majority (95.2%) reported at least one side effect. The most common side effect was injection site pain (91.8%), followed by fatigue (62.5%), headache (36.4%), and muscle pain (34.9%). The majority of local side effects occurred after both doses (74.4%), while most systemic side effects occurred after the second dose only (56.2%). Most local (94.2%) and systemic (93.3%) side effects resolved within three days after vaccination. Females participants’ adjusted odds ratio (AOR) showed they were 2.566 (CI 95%: 1.103–5.970) times more likely to experience post-vaccination side effects, and the participants who received two doses reported an increased AOR of 1.896 (0.708–5.077) for experiencing side effects. The results of this study imply that mRNA-based COVID-19 vaccines are highly probably safe for young adults, and further studies are required to investigate the role of medical anamnesis, prior COVID-19 infection, and gender in side effects incidence.