Fish,
Per DE law, you do not have to take the class before you jump through the other hoops. However, it may delay your permit slightly, should you be approved.
If you prefer to take your class after you do all the rest of the paperwork, I beleive it's the AG's office who sends you a letter (might be the Prothonotary's) that says, congrats, you've been approved, provided you provide a certificate you took and passed the course.
there are plenty of courses out there that should allow you to take the course almost immediately, but then you have to pass the course, and submit the certificate. As I mentioned, there is a slight delay, but maybe only by a few weeks at most.
Hope this helps,
Moosie
FYI, the permits in DE aren't as hard to get as one might think. If I were you, I'd apply.