So I wanted to stream my nintendo 3ds to twitch and start speedrunning some games. Are there any good ways to do that without spending money or if I do have to spend money what are some cheap options. Any help would be appreciated thanks.

If you got a new 3DS/2DS you can homebrew it and download a software that allows to stream it, google it for more infos about that. WARNING: Your system may get destroyed while doing this!!

The safer method is spending like 100€ on a capture card, this also works on the normal 2DS/3DS.

You can also use a camera (or phone) link that up usb to your computer to record then use a aux cable for sound. And less light. That's if you have no money. :) ^^

Any recommended websites to purchase capture cards off of?

Both http://www.3dsvideocapture.com/phone/ and http://www.3dscapture.com

Are the main places for purchasing 3DS capture cards. They are expensive though and there is no way around it. DS/3DS is unfortunately a pricey investment. There are homebrew ways to stream but there used to be questions about its accuracy and it had bugs. Maybe those have been fixed by now but I don’t know for sure.