Vaccinations are without doubt one of the greatest achievements of modern medicine, and there is hope that they can constitute a solution to halt the ongoing COVID-19 pandemic. However, the anti-vaccination movement is currently on the rise, spreading online misinformation about vaccine safety and causing a worrying reduction in vaccination rates worldwide. In this historical time, it is imperative to understand the reasons of vaccine hesitancy, and to find effective strategies to dismantle the rhetoric of anti-vaccination supporters. For this reason, we analyzed the behavior of anti-vaccination supporters on the platform Twitter. Here we identify that anti-vaccination supporters, in comparison with pro-vaccination supporters, share conspiracy theories and make use of emotional language. We demonstrate that anti-vaccination supporters are more engaged in discussions on Twitter and share their contents from a pull of strong influencers. We show that the movement's success relies on a strong sense of community, based on the contents produced by a small fraction of profiles, with the community at large serving as a sounding board for anti-vaccination discourse to circulate online. Our data demonstrate that Donald Trump, before his profile was suspended, was the main driver of vaccine misinformation on Twitter. Based on these results, we welcome policies that aim at halting the circulation of false information about vaccines by targeting the anti-vaccination community on Twitter. Based on our data, we also propose solutions to improve the communication strategy of health organizations and build a community of engaged influencers that support the dissemination of scientific insights, including issues related to vaccines and their safety.