As already noted above, UA and VN have an interline baggage agreement. That agreement applies when flights are on a single ticket. As jsloan explains, it is a different story when the flights are on separate tickets. While UA can check bags through to VN, it is against UA's published policies to do so on separate tickets because the two carriers are not alliance partners.

I will just add here that UA now has the most generous policy of any of the major US carriers in that it will still check bags through to *A partners on separate tix. AA and AS now expressly disclaim any obligation to do that, even when the second flight is on an alliance partner. DL will only do it with specific listed partners.