Then there is the cytokine storm to contend with, an overreaction of the immune system which causes serious inflammation.
at_239, it might help in understanding to know that "pneumonia" is a catchall term that just means an infection in the lungs. You don't actually get vaccinated against pneumonia per se, you get vaccinated against particular bacteria that can cause pneumonia. So, as ANM says, you are not protected from getting pneumonia from coronavirus, or from any other virus.
However, a person with pneumonia from a virus can be made even worse when they get a bacterial infection in the lungs on top of the virus infection. That's called a secondary infection. So yes, you are *somewhat* better off - though not in the way that you were originally asking about.
Severely ill patients from coronavirus are getting antibiotics against secondary infections, and it turns out that azithromycin also has antiviral activity.
No, unfortunately otherwise they would just vax us all for pneumonia. Sepsis is a leading cause of deaths from Covid.