.Students may not use the staff car park.

.Visitors may not feed the animals.

We can use may not in those occasions, but how about we only use "may"? Like:

.Students may use the staff car park.

.Visitors may feed the animals.

Are these two OK?

Thank you very much for your reply!
These are correct and common and natural.

Students may wish to use the restroom before boarding the bus. (Ouch! this is a different sense of "may." Guess I'm having a bad day.)

Students may refer to their textbooks during the exam.
In these sentences, may and may not are essentially opposites. Whenever you have this sense of may (have permission to), may not (do not have permission to) is also possible .
You may park here. = You have permission to park here.
You may not park here. = You do not have permission to park here.
may grants permission; may not denies it.