Ok, well I'm too lazy to try and fix the soldering iron right at the moment (and yes, I know it's been quite a few weeks since I received the help, but I'm
that lazy), so now I'm working on the contraption that it will shoot. I've drawn a diagram:
That's it when it's fully open. It'll be closed when it's inside the web shooter, and then it will open up when it gets shot out (this is a top down view, just to be clear).
So, here's how it's going to work. There will be a small square made out of wood (I don't know what it's called, but I mean the really light stuff that you can snap easily). Then at the end of each of those long bits, there will be a ring which fits onto each side of the square, as you can hopefully see in the diagram. And on either side of the rings I'll glue some other, smaller rings to stop them sliding along (that's what the black things are).
I'm going to make a prototype now, using card for the long bits instead of cork, which is what I'll use for the final model.
Also, to those two users who gave me advice about the soldering iron, thank you very much!

I think it's probably what the second person said, though, since the tip is completely black...