You checkin using the airline operating your flight. This means if using an app, use the VS app. Yes VS normally has its own checkin, even at JFK they have their own checkin positions. DL may man the positions with DL agents in VS uniforms at some locations.
DL currently has no PE product. C+ is simply a slightly elevated level of service in the main cabin. It's one of the reasons I feel DL misrepresents C+ as PE with the change to W fares.