You can substitute. Folks have successfully been making soap for hundreds and hundreds of years without DW. While the process of distillation has been around for thousands of years, the use of DW in soap making has only been something that is recent. And in researching commercial soap making and the benefits of using DW, I think I can say with almost certainty that they aren't using DW. If you have any doubts...think back to cleaning the bath tub before you started using artisan soap...I have definitely noticed the difference.
While DW is preferable because it doesn't contain any kind of contaminants...it's pretty much just 'pure' H20. But if you don't have access to DW (not everyone does), you can use other sources of water if that is all you have. And you don't have to use water, you can use milks, aloe vera juice and other liquids because the only reason why we use water in the first place really is to dissolve Sodium Hydroxide/Potassium. But if you do use other sources, you need to account for the contaminants. This can include running your Lye Solution through a coffee filter, fine cheese cloth or mesh screen to filter out bound contaminants and lowering your SuperFat.
ETA - I am not a science professional, but I can do the research.