The general take on shamans is that they borrow their powers from the elementals in a myriad of ways, but that seems to be an exception for the standard Zandalari trolls.
I suppose their powers would be connected to the Loa instead? But if so, the Loa makes kind of a ‘bridge’ between the Elemental beings and the shaman? Kind as if you’re hiring an accountant to do your taxes while the other shamans do their taxes themselves? 
(lol, sorry, poor comparison).
What could be the repercussions of a Zandalari shaman that decides to follow the Earthen Ring’s methods instead of their culture’s faith?
Thank you!
That doesn’t necessarily have to be the case. I highly doubt there’s anything in particular that would prevent a Zandalari Troll from evoking the elements the same as any Shaman. Their power could also be provided, or perhaps just somehow buffed, by the Loa they’re devoted to, though I don’t think it’d be a necessity; I imagine it would also depend on the Loa and their own abilities as, to me, some like Kimbul don’t make as much sense giving their followers elemental powers as someone like Akunda, but I digress. It’s not as if learning to use the elements would take away from their devotion. A Troll’s Loa could arguably be an Elemental as well, given what Loa technically are by definition. There are cases of personal or familial Loa in lore, in addition to those commonly worshipped.
I don’t think the Earthen Ring has any single enforced methodology of their own. They’re a collective of Shaman from different backgrounds who’re united by a common goal. Individual members have displayed here and there some differences in methods. There are Taunka Shaman among the Earthen Ring who, in accordance with their ways, make willfully force the elements into service, or Goblins who may treat the elements as tools to be used.
That’s a general rule…Lore has plenty of exceptions.
Also remember that classes are player mechanics, they do not define the lore, they abstract it.