The Japanese sauce term for the soy-based dip usually served with gyōza is tare sauce.
In Japanese cooking, tare is a broad word for a dipping, glazing, or seasoning sauce. Gyōza tare is commonly built around soy sauce, then sharpened with vinegar and sometimes enriched with chili oil, sesame oil, garlic, or other aromatics. The exact recipe varies by restaurant and region.
Tare is not the same thing as teriyaki sauce, although teriyaki is one kind of Japanese tare used mainly for glazing or marinating grilled foods. Tonkatsu sauce is a thicker, fruit-and-spice condiment, while ponzu is typically a citrus-seasoned sauce. Those distinctions make tare the best canonical answer here.
Japanese gyōza developed from Chinese jiaozi, but the Japanese version is often associated with thin wrappers, a garlicky filling, and pan-frying. The dipping sauce is usually served separately so diners can adjust the balance of salty, sour, and spicy flavors.