Yes. Although variations on the vaccine require less approval than a totally new one. This process takes place every year for flu vaccines.
Never for mRNA vaccines though - will be interesting to see how the administration/FDA handles this
While I know nothing about vaccine development I guess that a "release often in small batches" might be a better mode than doing mass clinical trials which are slow and costly. As far as I know it's possible from a manufacturing standpoint as the updates would mostly be changes in mRNA code. I think people not dying from being properly immunized outweigh the risk of a small number of people facing side effects if done properly.
Close to mid terms + if it becomes an actual big body count/hospital overflow in the US I'd bank on political pressure to approve fast.