Where would I look to find my WEP key?

What router do you have? 

WEP is pretty old and not secure. You should be at least be using wpa2. 

What video game console is it? 

Wireless Security: WEP, WPA, WPA2 and WPA3 Differences (techtarget.com)

The model says DG2470A. So I hope that helps there. I'm specifically trying to use a DS. While the official services are down, there have been a few proxy servers made for private games between small groups. To connect to my modem it asks for the WEP key, but the default security on it is indeed WPA2.

I wouldn’t use it with your current router. 

What I’d do is pick up an old router and connect it to your current one and just have the unsecure WEP on that older/sketchy router that you can power off when not using the ds. 

Alright, any cheap ones you may recommend? It'd only be for usage with my DS. Additionally, would I need anything else like any specific wires or such?

Just look in your old parts bin or on the local craigslist/offerup/facebook. Usually people are getting rid of devices that are older for pretty cheap. I wouldn’t pay much. 

You’d set the 2nd device up in access point mode with wep security (or even open for all the good wep does) and only power it on for when using the DS.