Understanding the factors that influence the intention to use vaccines is crucial for implementing effective public health policies. This study examined the impact of various cognitive, affective, normative, and sociodemographic variables on the intention to be vaccinated against COVID-19 with the first-generation AstraZeneca vaccine. A survey of 600 residents of Spain was used to assess the influence and hierarchy of the drivers of the intention to vaccinate via least-squares and quantile regressions. The most significant factors were the perceptions of efficacy and social influence, both of which had positive impacts (p < 0.0001). The positive influence of fear of COVID-19 and the negative influence of fear of the vaccine were also significant in shaping the central tendency toward vaccination. However, these fear-related variables, particularly the fear of COVID-19, lost importance in quantile adjustments outside the central tendency. Among the sociodemographic variables, only the negative impact of income was statistically significant. These results are valuable for the development of vaccination policies because they measure the sensitivity of attitudes toward vaccination to exogenous variables not only in the central values, as is common in similar studies, but also across the entire range of responses regarding the intention to vaccinate. This additional analysis, which is not commonly performed in studies on vaccine acceptance, allows us to distinguish between variables which are consistently related to the intention to vaccinate and those that influence only expected responses.