Hello there,
Thank you very much for reaching out to us. I'm sorry to hear that you are having trouble transferring items to your Vault. As viiTactiiCZz has stated, we've received reports of this issue and are investigating the underlying issue further. For any updates on this issue, we recommend viewing our Patch Notes as they become available, posted on the [url=https://www.bungie.net/7/en/News]News page[/url].
In the meantime, you can use the workaround mentioned (clicking the left thumbstick or the accompanying keybind (on PC)) to enter Vault Mode on the character screen. Alternatively, you can use the companion app.
They're aware of it, but haven't given any ETA on a fix date.
For now, you can trigger vault mode (to store items) by opening your character menu and pressing L3/Left Thumbstick if you're on console, I believe its CTRL if you're on PC.
Press the same button once you're done storing items to swap back to equip mode.