OK. I just fixed it and it's awesome. I attached the TPopupMenu to the TImageEnVect component through the PopupMenu property, which is what I had done before. I double-click the TImageEnVect's OnMouseDown event to create a method stub and I have the following code:
if Button = mbRight then
ImageEnVect1.MouseDown( mbLeft, Shift, X, Y );
And it works! The popup menu pops up with the right-click button and the element(s) under the right-click are selected! The TPopupMenu's items are all connected to TActions, so they show the appropriate actions based on what's currently selected (via the TAction.OnUpdate handlers).
That was too simple and way fun.