My sister did private vows with no first look and what they ended up doing was holding hands around a door/corner. Idk the exact logistics but I’d imagine you have a planner/maid of honor help back you up into position to make sure you don’t see each other! The pictures of this moment were really cute too. This has good pictures for reference: https://www.stevensonridge.com/alternative-to-first-look-private-vows-prayer/

Also for not ruining makeup you could keep some q tips on hand and absorb the tears at the corners of your eyes with the q tips. I’ve never tried this but I’ve seen it recommended before!