Trust in vaccines and in the institutions responsible for their management is a key asset in the global response to the COVID-19 pandemic. By means of a structured multi-scales survey based on the socio-cognitive model of trust, this study investigates the interplay of institutional trust, confidence in COVID-19 vaccines, information habits, personal motivations, and background beliefs on the pandemic in determining willingness to vaccinate in a sample of Italian respondents (N = 4096). We observe substantial trust in public institutions and a strong vaccination intention. Theory-driven structural equation analysis revealed what factors act as important predictors of willingness to vaccinate: trust in vaccine manufacturers (which in turn is supported by trust in regulators), collectivist goals, self-perceived knowledgeability, reliance on traditional media for information gathering, and trust in institutional and scientific sources. In contrast, vaccine hesitancy, while confined to a minority, is more prominent in less educated and less affluent respondents. These findings can inform institutional decisions on vaccine communication and vaccination campaigns.