Can you use a semicolon after an interjection/exclamation?

The usual form for using interjections involves following them with a comma, period, or exclamation point. For example...

"Well, I honestly have no idea."
"Dude! Where's my car!"
"Wow. I have nothing to say to that."

You get the idea. So! Can one use a semicolon after an interjection? I had never seen it done, prior to today, and I was curious.

I'd advise against it. The semicolon is conventionally used between independent sentences linked in thought, or in enumerations. Using them in the examples you gave would look rather unusual; I expect that you will not find many of those in educated writing.