Umh... Yea, is this greedy? And am I allowed to?

[Cranky Old Man]

Short answer: No, you can't get paid.
Long answer: Yes, you can get paid if you are a developer so award-winning that Frictional Games themselves are willing to co-develop the custom story with you, and presumably share the profits with you as well. So far this has happened once (with A Machine For Pigs). You'd have to speak to FG about it, but I believe that most developers would just be wasting their time doing so.
Here's an idea: If you're a less skilled developer, try to assemble other developers who would also like to make a buck, and pitch the idea to FG of making a sort of Humble Indie Bundle thing. Ten mediocre stories could be just as worth paying for as a great one. Then again, there are good stories out for free, so I don't think that there's a shortage of them.
Also, I don't trust developers who does it for the money.