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 spread 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 COVID-19 misinformation and vaccination hesitancy. Our study shows that only half of the online users exposed to rumors might have seen corresponding fact-checked information. Moreover, depending on the country, between 6% and 37% of individuals considered these rumors believable. A key finding of this research is that poorer regions were more susceptible to encountering and believing COVID-19 misinformation; countries with lower gross domestic product (GDP) per capita showed a substantially higher prevalence of misinformation. We discuss implications of our findings to public campaigns that proactively spread accurate information to countries that are more susceptible to the infodemic. We also defend that fact-checking platforms should prioritize claims that not only have wide exposure but are also perceived to be believable. Our findings give insights into how to successfully handle risk communication during the initial phase of a future pandemic.