When Jenner's vaccine - which was much safer and more acceptable than the earlier, expensive and risky practice of variolation - was recommended for mass use, there was widespread acceptance and within five years of his development of it in the late 18thC it was being used widely across Europe and within ten it was in global usage.
However, opposition to Jenner's vaccination was widespread and very active in Europe and North America. It came from a variety of angles. Some felt that using materials from cattle was unsanitary, others considered that the use of 'lower animals' was 'unchristian'. Some simply did not believe that smallpox was passed from person to person, while others quoted what we would now call 'human rights'. Throughout the 19thC there were armed riots, peaceful protests and marches on Town Halls and the like by large - for that time - groups of people in many different places.
There was also a great deal of fake news and publicity, not dissimilar to today's claims of tracking microchips, 5G, magnetism and 'shedding radiation', in which pamphlets were distributed to the increasingly-literate population of the 19thC, with titles like with titles like 'Vaccination: its fallacies and evils', 'Vaccination, a Curse' and 'Horrors of Vaccination'. Claims we would today recognise as nonsensical - that vaccinated people would develop bovine features, or that vaccinated women would roam the countryside 'at certain times' looking for a bull - were made, and there were many cartoons published - which even the illiterate could understand of course - depicting the evils of vaccination as seen in the fervent imagination of cartoonists of the time.
Governments widely tried to encourage the practice of vaccination; many countries made it free and many moved towards making it compulsory, which resulted in Anti-Vaccination Leagues being formed and protests becoming more organised. Here in the UK, a conscience clause was eventually added to vaccination legislation and things quietened down considerably, at least here.
Through a combination of quarantine, understanding of disease processes, vaccination and re-vaccination, smallpox outbreaks became increasingly rare in Europe and most of the Americas during the 20thC. However, even as late as 1959, over 2 million people were dying from smallpox every year mainly in Africa and Asia.
Of course, it wasn't 'just' vaccination which eradicated smallpox - it was the disease surveillance and containment methods developed by the Czech epidemiologist Karel Raška during and post WW2 which enabled the WHO's smallpox eradication campaign to be effective.
