Can I use any USB-C cable I already have with PowerLab C?

Using an ADInstruments branded cable is strongly recommended. Other cables may work but there are a number of different considerations here and the terminology/branding used in the industry is inconsistent and potentially confusing.
The PowerLab Event Link feature (PEL) of C Series devices require cables that support ‘USB-C Alternate Mode’. USB-C Alternate Mode is a part of the USB-C specification that allows non-USB data to share the USB-C Cable alongside normal USB traffic and power. The most common uses of this in the PC world are Thunderbolt (a high speed communication protocol for monitors, external drives etc) and Displayport (for connecting monitors). For this reason you will typically see cables that support PEL branded as ‘Thunderbolt’ or ‘Displayport over USB-C’ compatible.
USB-C Cables also come with a variety of different power ratings. ADInstruments branded USB-C to USB-C cables all support alternate mode and 100W or power.