Is it bad to workout the same day I got a flu vaccination? Isn't moving the muscle around a good thing?

No.Continue reg w/o. I see no reason to change your activity over a flu shot.I have gotten one yearly & had no issues working out the same day. Any work your body is doing recognizing the vaccine & developing immunity is done in the background by your immune system. If you do have dizziness or fever as can occur in some, a day to recoup would be reasonable.