While assembling a treadmill, I screwed through one of its wires (pretty sure it's the I/O cable). Can I fix this with electrical tape, or do I need to get this repaired? Any advice is much appreciated!
If you're absolutely sure you only damaged the outer insulation of an I/O cable then electrical tape would be Ok. If any other cable insulation or conductors are at all damaged, then you should get it repaired or get a replacement cable. Treadmills can be dangerous so you don't want to take chances with them malfunctioning or the dead man switch not working.
That appears to be a data cable.
data is normally low voltage and electrical tape would be fine starting 1/2” or more away make a tight wrap over coating at least 1/2 with each additional wrap will provide plenty of insulation for this.
The NEC doesn’t deal with equipment but having worked for a large company that was the method used to repair skinned cables.
If copper is showing on multiple conductors sealing then protecting with tape is preformed for field repairs.