I had the impression that the anti-vax movement started with Andrew Wakefield, a British doctor, because of the flap around 1998 when he published a paper in the Lancet which suggested that the Measles, Mumps, and Rubella vaccine (MMR) had not been properly tested and could cause autism. Usually publication in the Lancet means good research, but it turned out that he'd been paid to find out if there was evidence to support a legal case filed by parents who believed the vaccine had harmed their children. He invented the evidence to support his conclusion and his results couldn't be reproduced. By 2010 the British General Medical Council had ruled against Wakefield on several issues and the Lancet withdrew the paper. Wakefield is no longer allowed to practice medicine in Great Britain. This is just a summary, if you're interested in the Wakefield incident, read the linked article on him.
I got a lot of this from the History of Anti-Vaccination Movements, a 2018 article on the site History of Vaccines. I recommend the article to the interested. It reminded me that people have been objecting to vaccines since before vaccines existed as such (the concept was developed by Edward Jenner in 1798). The reasons aren't very different from what we're seeing now: people don't trust doctors, or the government. People don't like being told they have to do something. People are afraid vaccines will harm them.
A lot of people on social media have been referring to the general acceptance of the polio vaccines in the 1950s, in the United States, as the standard for public acceptance of vaccines against a horrible disease, and comparing it to current rejection of the COVID-19 vaccines. It was the exception. There were public objections to the smallpox vaccine, to the Diphtheria, Tetanus, and Pertussis (DTP) Vaccine, and of course to the Measles, Mumps, and Rubella (MMR) Vaccine. The only reason there were no objections to a vaccine for the 1918 influenza is because a vaccine was never developed. There were objections in 1918 to wearing masks.
Apparently people in the 1950s were simply more afraid of polio than they were of the vaccine, a reaction we haven't seen in the people refusing the COVID-19 vaccines. At least until they're in the ICU.