If my cucumber vine had a fungal infection, do I need to disinfect my trellis before using again, and if so, how?
If your cucumber vine had a fungal infection, the cause was probably too much rain and too acid of a soil. Fungi spores are everywhere and blow around like dust in the wind. Plants survive this by being healthy, if they're not healthy, they don't survive it. Rather than trying to fight the fungi, I would focus my energy into fixing the soil for next year by adding lime right now.
You can disinfect the trellis if you want, but I wouldn't see much use in it unless you can also disinfect the garden, your yard, the neighbor's yard and probably much of your county from where the fungus originated.