Transactions happen/get confirmed on the blockchain and not on the crypto wallets!
There is no need to keep BC Vault connected or even the application running once you have submitted the transaction to the blockchain or you expect funds to be received.
Exactly the same as with your banking account, you do not have to be logged into online banking system to be able to receive funds or for the send transactions to complete.
Let's consider two scenarios:
You are expecting crypto to be received to your BC Vault walletWhen someone is sending you crypto, this transaction is send to the blockchain and once it will be complete (confirmed) your wallet will be able to access this funds.
You are sending crypto from BC Vault walletOnce you have signed the transaction and submitted it to the blockchain for processing (You get: "Transaction successful" notice) everything else is done on the blockchain itself.