Yes it's perfectly fine to exercise after having vaccinations.
The exercise will help your body and boost your immune system as well.
I exercise daily and have done so even after having vaccinations including the flu shot vaccination and it's never been a problem.
The doctor told me it was okay to exercise after the vaccines but just wait at least 30 minutes after having the vaccination before doing too much strenuous exercise.
But walking, jogging etc are all fine.
Exercise is good for your health and will help you live a longer and healthier life.
The exercise can also boost your immune system and may even help prevent the cold and flu virus as well as prevent heart attacks, strokes etc.