Wallet transactions are all effectively presented as offline. I usually pay with Apple Wallet at Maverick (fuel/cstore). When I do so, the rewards account, which is also in Wallet, automagically applies. It stays even if I remove the app. I’ve seen it be broken in various ways over the past several years but at this point it all works right. The app matters in theory to authenticate and generate the Pass, but is otherwise inert. I had it running with no iOS permission enabled it but never used it and eventually dropped it. It had points and deals and such, and that’s what websites are for.