Both Firmware and Application versions will be checked by the application itself and when update is available, you will be notified and a simple click is enough to do it.
Do not worry about the Firmware update as BC Vault uses redundant partition for the update. If anything goes wrong (like connection drops, USB cable is pulled out or something else entirely) your BC Vault will still work.
WARNING: ALWAYS DO BACKUP TO MicroSD CARD AND/OR PAPER QR BEFORE UPDATES (just in case).