Just to clarify, what kind of orcs are these orcs? Faerun Orcs out of Volo's and Monster Manual? WarCraft orcs? Tolkien orcs/goblins/uruk-hai? Warhammer 40k orkz? Hipster orcs that try to buck assumptions of orc-dom by only being smug jerks that aren't 'evil'? Klingons? I ask because there's underlying lore stuff that could be used to justify a non-lethal outcome.

Absent any more information, maybe you could go for something where the pursuer has to prove they have the strength to keep the prey away from the challenger. If one or the other can keep their opponent far enough away from the prey for a certain amount of time, they proved they can keep the kill long enough for their followers to scavenge what they want anyway (not that this actually happens during the match; it keeps would be scavengers from fighting each other.) So maybe something like keeping the other X feet away from the prey for Y rounds. Ultimately, it's about claiming the prey rather than besting/ killing the other orc.

If you want to make this fun using orc traits, their Aggressive trait technically allows them to push each other extra far.* Grapple -> Move -> Aggressive allows an orc to push someone a full 30 feet away. Maybe have the challenge start with both parties grappled. Then they roll initiative to break the opponent's grapple so they can push the other. This kind of stacks the deck for an orc to win vs a non-orc, but a high athletics check, a shove at a pivotal time, or just knocking the other guy out can allow the non-orc to win.

*Some DMs may not read the grapple rules as allowing a push, but a game of orc bulldozers is fun to imagine, if only for this challenge.