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 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.
Interesting, in that case, it would make sense to have Native Segwit addresses in Bitpay wallet to save on the fee I am spending when I send transaction to another address?
Any idea if Bitpay supports Native segwit address in their wallet by default?
I checked and found that they added support in Bitpay wallet to allow sending to Native segwit addresses
https://bitpay.com/blog/bitpay-open-source-segwit/