Can a baby get the mmr shot a week before there 1st birthday instead of exactly on their birthday ? and then the second shot before school?

It would be up to the pediatrician giving the vaccine but getting a vaccine 1 week early should not be a problem. The pediatrician can then let you know when the next dose would be due. Good luck to you!

If memory serves me correctly, we used to give it at 9 months, about 3+ decades ago, but they moved it trying to get a longer lasting immunity from it. Eventually they added the 2nd dose at 4-5 in response to college age kids who occasionally had break through measles. The 2nd dose solved the breakthrough issue. I wouldn't worry about it.