I think that (at least where I live) the biggest problem is not will the shop implement payments through LN. Shop must have fiat for all sells so they have to automatically exchange BTC for fiat when they receive it. That's where BTC payment gateways come in place and I guess when they implement LN, shops will accept it too.
Most stores that accepts bitcoin(besides a small minority) accepts bitcoin in the first place anyway, whereas the bitcoin are automatically converted to fiat. Fortunately, open-source platforms like BTCPay exists for this exact purpose; and they've implemented LN too:
https://docs.btcpayserver.org/FAQ/FAQ-LightningNetwork/