When the threat of COVID-19 became widely acknowledged, many hoped that this pandemic would squash "the anti-vaccine movement". However, when vaccines started arriving in rich countries at the end of 2020, it appeared that vaccine hesitancy might be an issue even in the context of this major pandemic. Does it mean that the mobilization of vaccine-critical activists on social media is one of the main causes of this reticence to vaccinate against COVID-19? In this paper, we wish to contribute to current work on vaccine hesitancy during the COVID-19 pandemic by looking at one of the many mechanisms which can cause reticence towards vaccines: the capacity of vaccine-critical activists to influence a wider public on social media. We analyze the evolution of debates over the COVID-19 vaccine on the French Twittosphere, during two first years of the pandemic, with a particular attention to the spreading capacity of vaccine-critical websites. We address two main questions: 1) Did vaccine-critical contents gain ground during this period? 2) Who were the main actors in the diffusion of these contents? While debates over vaccines experienced a tremendous surge during this period, the share of vaccine-critical contents in these debates remains stable except for a limited number of short periods associated with specific events. Secondly, analyzing the community structure of the re-tweets hyper-graph, we reconstruct the mesoscale structure of the information flows, identifying and characterizing the major communities of users. We analyze their role in the information ecosystem: the largest right-wing community has a typical echo-chamber behavior collecting all the vaccine-critical tweets from outside and recirculating it inside the community. The smaller left-wing community is less permeable to vaccine-critical contents but, has a large capacity to spread it once adopted.