Well, if you don't have a capture card the best you can do is emulation or recording with an actual camera. Some game boards don't accept emulation, while others do, but be sure to check as to whether they have any requirements as to which emulator you're allowed to use. You can do screen capture with OBS Studio or X-Split, it's what everyone seems to use for running. If you want to get away without having to spend anything, emulation is likely the best that can be done.
The capture cards for the original DS aren't too bad considering that you can also do GBA with them, but from experience they aren't an easy DIY (those tiny flex cables are a pain) and don't capture sound on their own. The 3DS card market is a rather wild one from what I've seen, so you'll just have to wait and pounce on any opportunities you see. The prices on those are just as high as buying a new 3DS and the makers seem to run out of stock or vanish rather quickly. I don't think there are any cards made for the 2DS, but they may be working on those.