Bring 3/4 cup water, the chayote, and carrots to a boil in a medium saucepan. Add papaya, vinegar, chiles, sugar, and salt. Return to a boil. Reduce heat; simmer, partially covered, until chayote and carrots are tender, 20 to 25 minutes.
Remove from heat. Process one-third of mixture in a food processor; return to saucepan, and stir to combine. Salsa can be refrigerated up to 2 weeks; serve warm, cold, or at room temperature.