Please excuse my ignorance here, but I have very little idea of what I've gotten myself into. I need to extend the length of the ribbon on this membrane switch from the circuit board with wires for flexibility. The problem is, I have no idea what I'm looking for or where to look.
You didn't attach a picture.
If you just want wires on your membrane switch, then a through hole PCB connector can be used to solder wires to. It really depends on what type of flex cable you have. Type "flex" into the search engine at Digikey or other supplier to find more.
Measure the spacing between the pin holes of the connector. If there are a .1" centers you might be able to build your own extension cable from header connector parts like this, cut down to 3 terminals:
Use a servo extension cable, should be able to find them in any RC hobby store, not gonna be flat cable but it may be flexible enough. If that connector isn't completly fused shut you may be able to open it and tap the leads directly.