No, you can't see Illuminations from Cape May. There are some places around the Epcot resorts where you can see the high parts of Illuminations, but to really see the entire show, you need to be in World Showcase in Epcot. There is quite a bit of Illuminations that takes place at water level. It isn't just high fireworks.
You can take an Illuminations cruise, or dine in Japan or the UK and try for a table that provides a view of the show. Otherwise, just stake out a spot anywhere around world showcase at the railing to watch the show. Be careful if you decide to set up in a spot at the rail to be sure that no trees or other structures block your view either forward or up high. If you ask CM's around the area, they can show you the good spots.