I made a lawful good Blue Dragonborn Cleric of the Tempest Domain. I was going to use him in the Adventurers League, but none of the Forgotten Realms storm gods fit my character. Talos, the storm god, is Chaotic Evil, the exact opposite of my alignment. I thought that it would be cool to have a storm themed character, with the blue dragon's breath weapon and choosing the tempest domain, but it won't work out without a good aligned storm god.
You have some choices
Your cleric's and your god's alignment do not have to exactly align.
There is an Elven Chaotic Good deity, Aerdrie Faenya, whose domains are Tempest and Trickery. (Sword Coast Adventurer's Guide, p 23). CG god, Tempest domain, and your character embraces lawfulness in the cause of Good.
There is also Deep Sashelas, Chaotic Good Elf god of the Sea (PHB Appendix B). CG god, Tempest domain, and your character embraces lawfulness in the cause of Good.
- There is the neutral halfling goddess, Sheela Peryrol, whose domains are Nature and Tempest. (Sword Coast Adventurer's GUide, p 23). I think a dragonborn serving a goddess whose symbol is a flower is neat, but you may not care for that. Think about the back story for a moment: an encounter wherein your character has a revelation about nature, growing, storms, weather, etc. It might work.
- A better fit might be Valkur, Chaotic Good God of Tempest and War (Sword Coast Adventurer's Guide p. 21). CG god, Tempest domain, and your character embraces lawfulness in the cause of Good.
- Istishia, God of Water, N, Tempest. Symbol is a wave. (Sword Coast Adventurer's Guide, p. 21)
- Akadi, Goddess of Air. (N) Symbol is a cloud. (Sword Coast Adventurer's Guide, p. 21).
Alignment ... we’ve decided to restrict alignment choices to keep groups from becoming too self-destructive. You may not play characters with either the neutral evil or chaotic evil alignments in the D&D Adventurers League. However, you may play a character with the lawful evil alignment, but only if you are a member of either the Lords’ Alliance or Zhentarim factions.
Deities. Whether or not your character is a cleric, if they worship a deity, you may choose only from those deities listed in the Deities of the Forgotten Realms and Nonhuman Deities tables in the Player’s Handbook or any of the deities listed in the Sword Coast Adventurer’s Guide.
Note that the AL does not require a perfect match between deity and character alignment. The AL DM resource is silent on the matter, but the Adventure League FAQ, page 6 says:
Clerics must choose a deity, but needn’t have the same alignment as their chosen deity
A Sage Advice Ruling that may be helpful
Jeremy Crawford has this to say about deity/cleric alignment matchups. I'd recommend that you discuss this with your AL Dungeon Master.
Younes Kamal Eddine @Younes_KE
@JeremyECrawford @mikemearls as a Cleric can you choose a deity whose alignment differs from yours?
2:58 PM - Mar 21, 2016
Jeremy Crawford @JeremyECrawford
A cleric's alignment needn't match the alignment of the cleric's deity. #DnD https://twitter.com/younes_ke/status/712005570589552640
1:13 AM - Mar 29, 2016
For non AL games, DMG pages 11 and 13 present to the DM clerics receiving their divine powers from Forces and/or Philosophies. (An example given is a Paladin serving Justice rather than a deity ... ). That's going to be a choice a DM can make, though as I recall the AL set up is most frequently Fearun based and therefore deity based.
The Adventurer's League says:
Whether or not your character is a cleric, if they worship a deity, you may choose only from those deities listed in the Deities of the Forgotten Realms and Nonhuman Deities tables in the Player’s Handbook or any of the deities listed in the Sword Coast Adventurer’s Guide.
Looking in the back of the PHB, the only good aligned temepst deity who fits those criteria is Deep Sashelas, Chaotic Good Elf god of the Sea. It might require some creative back story, but by the strictest reading of the rules, that would seem to be your only option.
However, looking in the Sword Coast Adventurer's Guide, p 21: "Valkur, Northlander god of sailors; CG; Tempest, War." No dwarven god with tempest domain, according to the table on p.22. SCAG 23: "Aerdrie Feenya, goddess of the sky; CG; Tempest, Trickery" from the Elven pantheon.
This does seem to be a design weakness in Wizards' Adventurer League play to require a deity from the sources they list, but deliver a paucity of choices for Dragonborn or Human clerics (not to mention requiring the purchase of another book—SCAG) to give you more than one (odd) choice.
The most generic fit would be Valkur.
But as you are a dragonborn, let me give you and other google researchers the closest fit for draconic pantheon.
Lendys, the goddess of harsh judgement, is a neutral deity which as some temper in bringing justice. So this would be a good fit,
Otherwise, you could pray to Zotha, in an uncommon belief that he wasn't cast down into the Sea of Fallen Stars because he was bad but because he should be the one who take over the Tempest Domain. (It's practically a wildcard deity for weirdos.)