What would be the best way to ask for $8,000 to get A 3D printer + supplies to make a prototype of an invention I have?
It sounds like you are asking for the money to purchase the equipment, I think there is probably a better solution or two that you could try.
1) How about identifying Universities that could help you prototype and that you could 'borrow' not just 3D printing time from. Potentially getting some smart people, students, facilitators, lecturers, into your project could help grow it faster.
2) Rent time at a 3D printing space. I'm sure you are protective over your idea, but 3D printers are becoming increasingly common and perhaps there's a good way of getting this done locally to you which doesn't require a huge upfront investment.
So say now you have brought down the required investment to create the prototype down from $8k to $500, perhaps that opens up more options - extending an overdraft, the classic friends and family, etc.
I liked someone else's suggestion of doing a Kickstarter or Indiegogo, seems like a good move but they do take alot of effort, planning, and time. Probably more than the initial $500 you could raise yourself through other means.
Good luck. Matt