The Dragonflight Great Vault is located in Valdrakken at 58, 57 in the Valdrakken Bank. It’s on the right as you enter.
If you don’t like any of the rewards you can select from for the week in the Great Vault, you can instead choose to receive tokens which can be traded to Evantkis, the Token Exchange vendor.
To open your Great Vault after a weekly reset, you do need to actually visit the Great Vault but if you just want to check how many reward slots your eligible for, you can open Group Finder and click on Mythic+ Dungeons tab and then click on center menu button “Great Vault Rewards”.