Do you capitalize Major League Baseball?

No. Major league baseball is a general reference to league baseball, and major versus minor league.

compare
Major Baseball League, which is a professional organisation.

So, also note the capitalisation here:
Boston Red Sox are a Major Baseball League team.

The official organization's name is "Major League Baseball" often abbreviated MLB. So, in my mind and following examples, we should capitalize "Major League Baseball".

Oh - I'm a better grammarian than sports fan. I got them the wrong way round. Here's what I meant to say:

Major League Baseball is a professional organisation, and so capitalised.

compare
Major baseball league, which is a general reference to league baseball, and major versus minor league.

So, also note the capitalisation here:
Boston Red Sox are a Major League Baseball team.

(At least I'd remembered there was a team called the Boston Red Sox! )