Answer/Explanation:
There is no correct answer here! This is your opinion. However, in my opinion no, it should not be patented.
The issue is that we have to balance intellectual proprety rights (i.e. crediting whoever invented the vaccine) with serving the public and ensuring everyone has fair access to the vaccine.
If someone has a patent for their vaccine, it stops anyone else from using their technology to make it. This greatly restricts who will get access to the vaccine and will disproportionately affect disadvantaged communities/countries.
It is also worth pointing out that all of the research in the last 10 months into the virus has been a community effort, with lots of resource sharing. While a particular company might invent the vaccine first, they will have used much of the work from the broader scientific community to do so. Is it then fair for them to patent the vaccine?