Can RBF be done by a third (or second) party I've been thinking about the following usecase: 
could a retailer/merchant offer clients a "no bitcoin fees" option when paying with bitcoin? i.e. if something costs 1 BTC on the site, the user will pay ONLY 1 BTC, and the merchant would pay the tx fee.

As I understand RBF is a doublespend of the same utxo that, as it has a higher fee, invalidates the original tx simple for being confirmed first. 

Or would it be possible for someone else to bump up the fee without having to have access to the utxo owner's private keys (in order to sign the RBF transaction)

Thanks!
No, without knowing the private key it is not possible to create valid transaction and pay additional fee. There are some other complicated ways to force the transaction be included into a block, for example CPFP or creating non-final transaction ANYONE_CAN_PAY.

The simplest solution is seting the cost to 0.9998 BTC, user will pay 1.0 BTC with 0.0002 fee. This will produce the same result as in your example but without any additional actions.