Weird. In Oregon, if you go to the ODF&W site, they sell licenses, permits and tags as if you're buying items at Amazon. You put everything in a shopping cart and pay for it all when you check out. The receipt can be used as proof and you can keep it on the site under your account for as long as you'd like. If you want a printed version, you just print it off.
I printed mine because I don't always carry my phone (computer) but I do always carry my wallet and that's where I keep my printed "account" that includes my fishing license.
It really isn't an either or when it comes to the permits and licenses you purchase on the ODF&W site. If you want to pint it, there is no problem printing what you've purchased.