ANA and United have round-trip flights from Chicago (ORD) to Ho Chi Minh City (SGN) for $581-$591 with a connection. Flights August to October. No Weekends. Departures on Monday, Tuesday, Wednesday, and Thursday.
Note: Fare allows for a stopover in Japan for about +$120. If you want to spend time in both countries this may be a better option than booking separate tickets. You’ll want to pick “multi-city” itinerary and select ORD-NRT/NRT-SGN/SGN-ORD as your route. Example screenshot is provided in the full blog-post.