I don't believe most old ones, Fender rev/trem for example, really require the shielding as the cable doesn't have any audio signal running through it. it's just easier to use a shielded two conductor cable than find a three conductor cable.
Actually, the shield is vital for the reverb footswitch on an old-style (BF/SF) Fender amp. So much so, that the stock Fender footswitch cable has a shielded wire for the reverb, with the wire for tremolo on/off running outside the shield. They do share a common ground (through the shield) inside the footswitch housing, but the shield is definitely needed. I once tried replacing the stock (stiff and too short) cable on a SF Twin Reverb using regular mike cable - using one each of the inner conductors for reverb and tremolo, respectively, and the shield as the common ground. I was then greeted by a loud "pop" each time I turned the tremolo on or off... For that one, I ended up using a dual shielded cable (basically a long stereo RCA-RCA cable), to separate the reverb footswitch line from the tremolo. The Twin I have now didn't come with a switch, so I built a small one just for tremolo on/off (leaving the reverb switch connector empty and the reverb constantly on). For that switch, I used non-shielded cable, which worked fine. But the reverb switch definitely needs a shielded cable.
But going back to the OP's question, most footswitches for (reasonably) modern amps simply control relays, rather than the signal directly, and will work fine with non-shielded cable. But there's no harm in using shielded cable, of course. And asking the manufacurer is always a good idea too