Monorails start running by 7am, if not earlier.
But the problem is that if you're at the Beach Club you won't be able to *get* to the Epcot monorail until Epcot opens at 9am (or 8am on an EMH Morning). This will also use up a day on your ticket and force you to stick with Epcot for the day if you don't have a Park Hopper.
Your best bet is to go out to the BC bus stop and get on either a "Character Breakfast" bus or a Magic Kingdom bus, whichever comes first. If the Character Breakfast bus, just tell the driver where you're headed and they'll get you to where you need to go.
-Rob