The COVID-19 pandemic has been damaging to the lives of people all around the world. Accompanied by the pandemic is an infodemic, an abundant and uncontrolled spreading of potentially harmful misinformation. The infodemic may severely change the pandemic's course by interfering with public health interventions such as wearing masks, social distancing, and vaccination. In particular, the impact of the infodemic on vaccination is critical because it holds the key to reverting to pre-pandemic normalcy. This paper presents findings from a global survey on the extent of worldwide exposure to the COVID-19 infodemic, assesses different populations' susceptibility to false claims, and analyzes its association with vaccine acceptance. Based on responses gathered from over 18,400 individuals from 40 countries, we find a strong association between perceived believability of misinformation and vaccination hesitancy. Additionally, our study shows that only half of the online users exposed to rumors might have seen the fact-checked information. Moreover, depending on the country, between 6% and 37% of individuals considered these rumors believable. Our survey also shows that poorer regions are more susceptible to encountering and believing COVID-19 misinformation. We discuss implications of our findings on public campaigns that proactively spread accurate information to countries that are more susceptible to the infodemic. We also highlight fact-checking platforms' role in better identifying and prioritizing claims that are perceived to be believable and have wide exposure. Our findings give insights into better handling of risk communication during the initial phase of a future pandemic.