Vaccine refusal can lead to outbreaks of previously eradicated diseases and is an increasing problem worldwide. Vaccinating decisions exemplify a complex, coupled system where vaccinating behavior and disease dynamics influence one another. Complex systems often exhibit characteristic dynamics near a tipping point to a new dynamical regime. For instance, critical slowing down -- the tendency for a system to start `wobbling'-- can increase close to a tipping point. We used a linear support vector machine to classify the sentiment of geo-located United States and California tweets concerning measles vaccination from 2011 to 2016. We also extracted data on internet searches on measles from Google Trends. We found evidence for critical slowing down in both datasets in the years before and after the 2014-15 Disneyland, California measles outbreak, suggesting that the population approached a tipping point corresponding to widespread vaccine refusal, but then receded from the tipping point in the face of the outbreak. A differential equation model of coupled behaviour-disease dynamics is shown to illustrate the same patterns. We conclude that studying critical phenomena in online social media data can help us develop analytical tools based on dynamical systems theory to identify populations at heightened risk of widespread vaccine refusal.