Since Native Segwit addresses (Bech32) use lower fee as compared to Segwit addresses, then would it make sense to ask the online merchants to provide a Native Segwit address for accepting payment instead of a Segwit address?
For you as the customer it doesn't really matter. The lower fee only applies to the sender, so if you send from a native SegWit address you save on fees no matter whether you send to another Bech32 address, a P2SH SegWit address or a legacy address.