We present VaccinItaly, a project which monitors Italian online conversations around vaccines, on Twitter and Facebook. We describe the ongoing data collection, which follows the SARS-CoV-2 vaccination campaign roll-out in Italy and we provide public access to the data collected. We show results from a preliminary analysis of the spread of low- and high-credibility news shared alongside vaccine-related conversations on both social media platforms. We also investigate the content of most popular YouTube videos and encounter several cases of harmful and misleading content about vaccines. Finally, we geolocate Twitter users who discuss vaccines and correlate their activity with open data statistics on vaccine uptake. We make up-to-date results available to the public through an interactive online dashboard associated with the project. The goal of our project is to gain further understanding of the interplay between the public discourse on online social media and the dynamics of vaccine uptake in the real world.