Can I change my name to my married name with social security now that I have filed my 2020 returns as married or do I need to wait until the returns are accepted?

It goes the other way around.  You don't have to change it at Social Security.  When you file your return you use whatever name is currently at Social Security.  What name did you use on your return?

I used my maiden name which is what all my w2s are under as well as my bank. But I need to change my name legally still so I was wondering if I started that process now will it impact my return? Is it best to wait or will it even matter?

what year were you married and did you contact the Social Security Administration to change your name?

the law allows you to use your maiden name even on a joint return but then you cant change it with the SSA. 

you may want to contact THE SSA if you haven't done so already.