This study focuses on factors that shape vaccine attitudes and behaviours in the context of a low-trust society. Our analysis focuses on the Polish vaccination programme against COVID-19, primarily on (1) the evaluation of the information campaign, (2) trust in the institutions, (3) trust in other people, (4) attitudes toward vaccine safety and efficacy, (5) attitudes toward restrictions related to vaccination (e.g., restricted access to certain services for unvaccinated persons) and the introduction of mandatory vaccination, (6) the evaluation of the government’s actions during the pandemic, and (7) political preferences. The study was conducted with a sample of 1143 adult residents in Poland (CATI). The explanation of the factors determining the COVID-19 vaccine was based on structural equation modelling (SEM). The model showed that the declared fact of vaccination was largely determined by a positive attitude toward restrictions related to vaccination and trust in vaccines. The formation of the provaccine attitude was to an extent determined by the assessment of the government’s campaign and actions during pandemic. While institutional trust had a positive effect on support for the ruling coalition (0.56), the latter on its own had the opposite effect (−0.61) on the formation of provaccine attitude. In the group who both trust institutions and support the parties currently in power, there are more of those who simultaneously reject the restrictions and mandatory vaccination and remain sceptical about the safety and efficacy of COVID-19 vaccines than those who both trust in the vaccine safety and efficacy and accept the restrictions and mandatory vaccination. This indicates that in the context of strong political polarisation, ideological affiliations may play a greater role in shaping vaccine attitudes and behaviours than institutional trust.