Mistrust in COVID-19 vaccines may hinder vaccination campaigns. We looked at cognitive determinants of vaccination intentions against COVID-19. We were interested in (i) the effects of stress and (ii) the effects of self-protection systems on attitudes and intentions to get COVID-19 vaccines. We conducted an online observational pilot study with 203 participants and used self-report questionnaires to assess perceived stress and vulnerability to disease, beliefs about a dangerous world, pandemic-related stressors, living conditions, attitudes and intentions toward the vaccines and trust in government management of the COVID-19 pandemic. The participants reporting high levels of trust in government and high levels of stress were more likely to have positive attitudes toward COVID-19 vaccines, although these two effects are at least partially independent of each other. We discuss how to improve the communication around COVID-19 vaccine policies.