Update: Registration is now open. (It opened Jun 28, 2021.) However, seats are filling up quickly, and many students are unable to register for their preferred location or date. Come on, College Board! You need to try harder!
Short answer: You should be able to register for the August 28, 2021 SAT ‘by early July’, which means before early July, so technically, any time before ‘early July’.
You can double check the SAT test dates page to verify yourself. If you do, you’ll see this:
August SAT registration (more information)
You would think that one would be able to register for the August SAT right now. I mean, it’s only single-digit weeks away (as of today, 2021-05-31, when I’m writing this). And all the other SAT dates were available for registration far ahead of time.
However, this year is quite different. We’re in the COVID-19 era, the year+ of cancellations, of flying out of state to other test centers, of showing up to a closed test site. So you can’t take anything for granted these days.
Last year (2020), registration for the August 2020 SAT opened up in late May. But College Board was scrambling a bit, trying to figure out how to accommodate rising seniors who needed to finish their SAT for their college applications. (It didn’t go well, as you can imagine.)