The Old Ways
Just as every druid race has a different type of druidism culturally, the same applies to shaman races. It’s different by culture. And as for worgen and Gilneans, the closest would be Harvest Witches. An animistic pagan belief described as “crude forms of druidism and elemental shamanism” (Chronicle 1), which frequently references the spirits of wind and earth in quest text.
“The earth speaks to us, . It will not abandon us even in these perilous times.”
-Celestine of the Harvest, Moonfire(worgen)
Once bitten, a Harvest Witch finds their connection with nature amplified, thus why they’ve often gone on to learn Cenarion druidism.
Sabrina Pilgrim
While worgen ingame can’t be Shamans, there are a few who come close. Apart from Harvest Witches, there is also Sabrina Pilgrim in Ashvenvale, who works with four elemental lords, and is described as “trying to help the Earthen Ring”.
“The elementals here are now my allies. You must do as they ask if we are to prevent the destruction of the rest of Ashenvale.”
-Sabrina Pilgrim, Go with the Flow
“It is done. The power of fire is subdued and the four elements are returned to harmony.”
-Sabrina Pilgrim, Vortex
Goldrinn
The wolf god is the worgen’s patron spirit, and is indeed a big possible connection between them and shamanism, in various ways. Being t
First and foremost, as you mentioned, spirit wolves. Shamans ingame can take on ghostwolf form, which is much like the druidic lore ability to use Pack Form. Thrall and Akama are known to summon spirit wolves, and so have worgen. See Wolf Master Nandos from Shadowfang Keep.
Goldrinn has been recognized as an important figure in Orc culture on Draenor. Which is likely possible, since druids still had connection to the Emerald Dream on Draenor, and because one of Draenor’s gods tried to take over the Emerald Dream. https://wow.gamepedia.com/Anzu
The final icing on the cake, humans cannot sense auras, but worgen can. This is stated in the novel Wolfheart, when Genn boasts to Varian that the worgen curse did more than enhance their physical senses.
“Your quarry? You jest!” Genn sniffed at his opponent. “You think you can take him from me? Listen to me, Varian Wrynn! The curse more than heightened our senses. We see things that no normal human can. Some call you Lo’Gosh, though that they use a Taur-ahe title for you I find ironic. Still, it is but another name for Goldrinn, as we have come to know our patron spirit since our transformation. I saw the aura of that spirit around you the first moment you arrived at the banquet, and even though you gave every indication of crushing our hopes then, I still held out for our chances because I could see his touch upon you as if it were your own skin. . . .”
Wolfheart pg 267
So in conclusion, we have an example of cultural interest in elements, a couple she-worgen who speak of and interact with elements, worgen who summon spirit wolves, all worgen who can now see auras of spirits (important for communicating with elemental spirits), and a connection to a God that has long since been involved in shamanism. All in all, a worgen shaman should be possible. But not common enough to be added as a class mechanically.