We booked a flight via Travelocity for my stepson to return home to Vietnam in October 2019. He will travel from MCI to SEA and then another Delta flight from SEA to ICN and finally from ICN to SGN on a Delta flight operated by Korean Air. Will he be able to check his luggage all the way through to his final destination in Ho Chi Minh City or will he have to retrieve his luggage in Seattle or Seoul and recheck through customs?
Generally, if it's all a single booking you can expect baggage to be checked through so you won't see it at layover point (except in some cases of international-to-domestic transfers, but there are none in the itinerary you describe).
Some ticket resellers will sometimes issue tickets spread over several bookings at the airline level even of you buy them at the same time from their website. This practice is generally considered rather dodgy unless the passenger is told exactly what is going on (because it also influences you rights in case you're delayed and miss a connecting flight), and I don't think Travelocity is one of them. If in doubt, look for small print that warns about connections not being guaranteed, or being "guaranteed" by someone who is not the airline.
In your case, it looks very likely that it is all a single booking -- especially the fact that the last leg is booked as a Delta codeshare rather than under its Korean Air flight number. There would be little reason to do that except to keep the entire itinerary on a single Delta ticket.