Sarah Illes Johnston differenciates the prophetic nature of divination that comes from prophecies and dreams to the ones like dice-casting and ornithomancy. Although both can be divine-inspired, the former two are naturally developed whilst the other ones must be learned. As a polytheist, knowing the divinatory ways associated with your deity can be quite a nice way to improve your relationship with said deity.
That being said, I'll list some gods here that were associated with divination methods, oracular shrines and whatnot:
Apollon, being the God of Prophecy, is related to most divinatory arts. The Oracle of Delphi was said to be able to make prophecies inspired by Him. Also associated with: Dice-casting (cleromancy), bird-watching (ornithomancy/augury).
Hermes, being the God of travels and roads is most associated with many aspects of those who travelled throughout the world - Seers being some of those. In his Homeric Hymn he is said to have received the power of divination from Apollon himself. He's particularly associated with astralagoi/psephoi - the alphabetic oracle, which uses dice-casting techniques (cleromancy) to predict the future.
Helios, the Sun-God had association with divinatory practices, as said by Apollonius, who uses a form of pyromancy (fire-based divination) after offering frankincense to Him.
Zeus, the King of the Gods, is said to have near all-knowing wisdom, with some saying that Apollon's prophetic power is essentially based on delivering Zeus' designs and divine will to the Cosmos. Zeus had a whole bunch of oracles associated with Him and in many moments of the Illiad where Calchas, the Seer is present, he predicts the future by usage of birds omens (ornithomancy) that deliver messages from Zeus.
Hades and Persephone, the King and Queen of the Underworld, were both honoured in the Nekromanteion - a temple structure that housed many divinatory practices regarding the dead (necromancy). Spirits, daimons and other chthonic gods were mostly associated with this form of divination.
Poseidon, the God of the Seas, also had association with oracles. Pausanias mostly relates him to the one at Delphi before Apollon claiming it as His own. Like many gods here listed, He is said to send omens in the sacrifices offered to Him, particularly bulls.
Selene, the Moon-Goddess also had a shrine and oracle described by Pausanias. The Goddess would answer whatever they wanted to learn through dreams, said shrine was also shared with her brother, Helios.
Many, many other Gods had their own ways of sending omens, communicating or helping people to read the future. Those are just some of Them in a list that I hope you guys find useful.
Oh well, I can't stress enough how important divination was and still is to worshipping the Theoi. Learning about it can be intimidating at first, but I hardly believe that you gonna regret reading about it.
Sources: Pausanias' Description of Greece; Ancient Greek Divination by Sarah Illes Jhonston; Greek Folk Religion by Nilsson; The Homeric Hymns.
