Really since there aren't live viruses or pathogens in the covid vaccines, they shouldn't warrant the same kind of caution in terms of exercise as the actual disease or even a flu or other milder respiratory illness. Just because one might feel sick after a vaccine doesn't mean one actually is.
That being said I was completely wiped out after my vaccine. In a few hours after the jab I had gotten a high fever and felt like I'd contracted a bad case of influenza. Took me the night and whole next day to recover so wouldn't have been able to exercise even had I wanted to. Once the symptoms started tapering off they went away in a couple of hours leaving only fatigue from the restless feverish night.