I doubt it will ever be required for the broader software field, but I could imagine a PE being required to write software for medical devices that can harm people, or power plants, or cars, etc.
I also think that many firms that design consumer products aren't run by PEs. So it may not be a universal requirement.
In my opinion the most analogous type of engineering to the average web dev shop is HVAC. Every building needs someone to design the HVAC system, but it's pretty routine work. There explicitly is a PE exam for that type of work, which seems like overkill to me, even though I'm very much in favor of PE exams.