As of now, all Mexican Nationals that I know have not been able to cross the border for over one year. I just heard from parents yesterday at practice that kids will not be returning to school in the fall because of vaccine shortages. Lots of people in TJ are vaccinated, but not so many in rural areas and places far from the US border.
Most of the people crossing the border now are vaccinated Mexican-Americans for work. I crossed last Saturday in a standard lane in about 30 minutes-it would normally take 3 hours.
There are still travel bans in place including IIRC against the UK where their citizens can't get in to the US. I could be entirely wrong about this but I don't think Mexico is on the ban (some quick research didn't show it on the list, but I don't really care enough to do a deep dive)....I think they are prohibited by the loose shut downs of the northern and southern borders Though, stupidly, if you cross illegally, the same prohibition doesn't seem to apply, even if you test COVID positive (I'm not into the weeds on the details here, but I recall there being a few news stories in Texas about COVID positive people being released by the migra in the last couple of weeks). More stupidly, US citizens can go to some of the banned countries and then return (my son's favorite bloggers Tara and Neil are on some holiday in Italy, for example).