In general, the U.S. constitution provides broad freedom of speech. This includes faculty, but there is no special academic freedom here - just the general freedom that all Americans (and many non-Americans) enjoy.
The political activities of many governmental employees are regulated by the Hatch Act. The Hatch Act is generally applicable (with exceptions) to many categories of federal employees, including people who receive only a small part of their pay from federal employment. Federal agencies like the National Science Foundation, Department of Energy, and others often employ faculty members in various capacities. In these situations, they would be bound by the Hatch Act.
What does the Hatch Act limit? It depends on the kind of position. This page summarizes what kinds of activities are prohibited.
None of this is directly applicable to the people who run American universities, because it is rare for those universities to be run by the federal government. I know of one (Haskell University is operated by the Department of the Interior to provide education for native americans). There are likely a few others, but not many.
Universities (like any other organization) have the ability to decide what kinds of political activity are appropriate for their official operations. They may choose to limit their employees in ways similar to the Hatch Act, impose their own rules, or do nothing at all. At least in my experience, most universities tend toward the latter end of that spectrum.
Outside of their official capacity, faculty (including administrators) may publicly support anyone they wish. However, there may be practical consequences for doing so (such as hurt relationships with donors or their board).