[quote]Hello there,
Thank you for reaching out.
We have been alerted to an issue impacting the Vault. At this time, we're looking into the cause of this.
In the meantime, you may be able to use the Destiny Companion app to access the vault. Your patience and understanding while this is ongoing is appreciated.[/quote]
https://www.bungie.net/en/Forums/Post/265024392
As for in-game, if you are in your character menu or you open your character menu while in the vault, if you look at the bottom right there will be a 'toggle: vault' with a button next to it. Using that button should swap you to vault mode and allow you to store your items.
If it doesn't work for you then you'll have to use an app (as Bungie mentioned above) to store them until a fix is pushed.