Is "while" considered a duration such that even if the card with the effect leaves play, the effect holds? Specifically on Letter of Marque:

Encouraged Assault – Reveal: Choose an opponent. While your total initiative is less than the chosen opponent’s, they are your rival.

Does Letter need to stick around for the rivalry to hold?

Short Answer: No, "while" is not considered a duration.

Examples of Duration:
For this challenge
For this game (Same as For this Challenge)
For this round
For this turn

A game consists of a maximum of 5 matches, a Match consists of a maximum of 5 rounds, a round has two turns (one for each player).

Letter of Marque will need to be in play in order for it to grant a rivalry.