The College Board has given several reasons for why it doesn't administer an SAT in summer. These reasons include concern about interest (how many students would actually sign up for a summer SAT), how it would affect the current SAT dates (would they add another date or have to move around one of the currently existing test dates), getting staff/testing centers (since schools are closed), and how it would affect students with fee waivers.
Of those reasons, I think the most weighty ones are probably the lack of staffing and test center availability over the summer (a lot of proctors are teachers, who during summer break may have other jobs/time commitments) and the fact that the College Board isn't sure how many students would actually sign up for a summer SAT (since over the summer students are often busy with summer programs, camp, work, volunteering, or even doing summer work for school). But that's just my best guess - I am not able to say for sure what the "real" reason is, since I am not part of College Board.