I was wondering if there's any. I know it's a weird question but please bear with me; assume you get a group of people ready to pay top dollar for some suits of plate armour. Thing is they are all vegan so they want nothing of animal origin. What would you use for straps?
I have some close people who are into medieval stuff but follow this philosophy, always see them checking food, clothing, cleaning products etc to make sure they have nothing of animal origin, so it got me thinking of alternatives.

Cloth webbing works for the military.

As far as I know it's not historical, but covering a stronger material with a facing of velvet or silk (faux silk if they have issue with natural silk, I'm not sure of vegan views on that) to match the kit could look nice.

Leather is hard to get away from when looking for period solutions.

However, for strapping, strips of canvas or linen folded several times and sewn well would work. It may require grommeting or eyeletting for extra strength or tear resistance.

Braided or twisted ropes might work as well, if we're talking straps used as hand-holds and such. If you needed a flat strap, then fabric is about it.

A flat chain is also an option, but it would be expensive and could well be more prone to causing chafing or blisters.