To answer the question that I'm sure is burning in everybody's mind - YES, vaccines can be patented.
One example of a vaccine patent is US3132073A which was granted in 1964 and expired in 1981. The patent was was filed and owned by Eli Lilly and Co, still a mega pharmaceutical company today. During its term, the patent protected a method of making an improved measles vaccine that has reduced side-effects than previous measles vaccines.
Part of the novelty and improvement here is that the vaccine is made by growing the measles virus in a chick embryo tissue sample instead of monkey-tissue. This avoids the problem of simian viruses that are associated with those vaccines that were developed with monkey-tissue.
Companies like GlaxoSmithKlein hold numerous patents relating to flu vaccines. In some cases, the vaccine composition is patented: https://patents.google.com/patent/EP2422810B1/. In other cases, the method of administering the vaccine is patented: https://patents.google.com/patent/US9730999B2/en. And in other cases, the method of manufacturing the vaccine is patented: https://patents.google.com/patent/US10398771B2/en.
Let's hope that someone finds a vaccine to COVID-19 sooner than later. Until then, stay healthy!