Vaccination against SARS-CoV-2 is postulated as the most effective measure to control the COVID-19 pandemic. However, the use of other protection measures is necessary to efficiently combat the spread of the virus. The aim of the present study was to determine the attitudes and intentions toward COVID-19 vaccination among non-regular social media users in Spain and to analyze how these factors could condition the acceptance of other personal protective measures once an individual has received the COVID-19 vaccine. A cross-sectional design was used in this work. In total, 719 subjects, ≥18 years old and of both sexes, were recruited from primary public healthcare centers to self-complete a questionnaire between March and April 2021. The majority of participants had a positive attitude toward vaccination and showed high levels of intention to be vaccinated. Likewise, except those participants who considered the vaccine to be the most effective measure to fight the COVID-19 pandemic, the rest of the participants highlighted the importance of continuing to limit social interactions and/or wearing masks even after being vaccinated. Since vaccination can create a perception of total immunity against SARS-CoV-2, it is necessary that healthcare staff organize effective awareness campaigns on the importance of maintaining personal protective measures until vaccination coverage is greater.