We just input our phone number for balance rewards while the cashier is scanning the items and then use Apple Pay. Never had a balance rewards card, so I would have no idea how to add one to Wallet. If this does balance rewards and Apple Pay at the same time (waiting for someone to field test it and report back), even better.
It used to be that all of our local Walgreens in Houston would require you to input your pin after using Apple Pay no matter the dollar amount, but that just stopped recently. Weird, but now much more convenient.
https://www.walgreens.com/balancerewards/balance-rewards.jsp
What I did was enter the phone number I had been using in the store to sign up with on their site. Once I had my account created, I downloaded the Walgreens app and added my account info there. Inside the app there's an option inside to add the Balance Rewards "card" to your Passbook.