If I own a game and the console is an emulator still illegal?

I'd like to stream Tomodachi Life but I don't have a capture card so could I get an emulator for a 3DS and play on there legally?

All emulators are perfectly legal. Having a ROM on your computer that you do not already own is not. So yes if you own the real game, it is ok.

Actual emulation is legal. There's is nothing wrong with one machine emulating another machine, it just when you start stealing software or disregarding licensing agreements things start to get a little dicey.