Bittrex is an exchange, it is a custodial exchange, which means all funds sent to your bittrex account or your friend's bittrex account is not yours on blockchain, bittrex owns all the coins on blockchain because they have and control the private keys to the bittrex addresses, which makes them to have the full control over all funds sent to bittrex addresses. So, why leaving your coins on a platform you do not have access to the private key? The best is to move all coins on custodial exchanges back into your noncustodial wallet which you have complete control over and protect its seed phrase and the wallet.
If you do not use the wallet that will give the option of username, you will have to tell bittrex customer care about that. Telling them you can sign a message from the address(es) used to make payment into your exchange account. If you use wallet like Electrum, let them know about that and how it is not named your name. Let them know how you can be able to sign a message with the addresses used to send coins to the exchange account which was already stated above.
But never leave funds on exchanges, move all yours coins to a noncustododial wallet to have full control over your coins.
“Is there a screen in your wallet program that shows your wallet address associated with your name or email address or just a list of your wallet addresses?”
Importing the seed on electrum wallet and naming the wallet your name could do that. But why should a wallet require email, noncustododial wallets requires no email. Tell them you can sign a message using the address is the best option to go for. Also never reveal your seed phrase or private key to anyone, be careful of wallet screenshot to never screenshot your seed phrase or private key.