Divination is one of the optional courses that students can start taking in their third year, along with Muggle Studies, Arithmancy, Ancient Runes, and Care of Magical Creatures.

The number of OWLs taken by Percy Weasley, 12, which is the most possible, suggests that he took Divination and got an OWL.

Academic Courses at Hogwarts (does not include Flying)
1. Divination
2. Muggle Studies
3. Arithmany
4. Ancient Runes
5. Care of Magical Creatures
6. Potions
7. Defense
8. Charms
9. Transfiguration
10. Herbology
11. History of Magic
12. Astronomy

This suggests that it is possible to at least take the OWL and presumably, NEWT-level Divination also exists.

I'm not sure where Firenze went during DH because he could not return to the Forbidden Forest either, Bane had banished him. Presumably Trewlaney kept teaching it.