Like in many countries, the French health authorities are pinning their hopes on vaccination to stop the spread of the COVID-19 pandemic. However, this requires the public to get the vaccine. Based on six COCONEL surveys carried out during and after the lockdown among the general French adult population, which aimed to explore life conditions during this period and individual perceptions of the situation, this paper aims to investigate the acceptability of a putative vaccine against COVID-19 among the French population, and more specifically to compare attitudes among the elderly, who are more vulnerable to the disease, with those of younger people. Out of the 8,036 respondents, 27.5% declared that they would refuse the vaccine if it were available. Among the age groups of particular interest, 29.2% of people aged 18–64 said they would refuse the vaccine, compared to 12.4% of respondents aged 65 or over. For both groups, this refusal rate fluctuated depending on the date of the survey, gender, household composition, level of household income, and uptake of the last flu vaccine. In conclusion, more attention should be paid to elderly people, as they were more likely to report hostility toward vaccination in general. Finally, considering that several vaccines are now available, further studies should be done to study vaccine uptake in relation to the vaccine offered.