It depends on how much funding we’re talking about here… you would probably want to find an institution-affiliated researcher (affiliated with a university or NASA or whoever), and sit down and talk with them about your idea. You don’t need to have “formed a hypothesis” to do this, just be able to discuss your idea. If anything does come of it, the researcher will want to write the proposal anyway. Be prepared for your idea to have been thought of and rejected in the past, or for it to be unfeasible. But anything is possible.
It’s unlikely that anyone will answer an unsolicited email asking for a meeting like this; there are a lot of cranks out there who think that they are undiscovered geniuses. And researchers barely have time to answer their own students’ emails.
One good way to make contact might be to attend a conference on relevant topics, and find someone giving a talk on a subject close to yours. Put up your hand after the talk, and say you’d like to talk to the speaker about an idea afterwards, if he is open to that. Conference attendees are kind of sitting ducks in this way, and they have time to kill, so it can be easier to approach them. Be warned, though – conferences are expensive.
If you have no science background, it’s unlikely that you would get to do any of the actual work on the project, but you could end up a co-author on the paper or something. Be sure to discuss your expectations about this up front, because authorship depends on a lot of factors that you probably aren’t aware of. And know that there isn’t going to be any money in it for you (i.e., for either of you).