I'm in Praiano (between Positano & Amalfi) now and have looked into finding SUP rentals. Bad news: there don't seem to be any.
I inquired here in Praiano at the dive shop and at a kayak rental/hotel store in Positano but turned up nothing. There are some wind surfers about and I think if you were desperate you could try to use one of those boards with a long paddle--I saw some kids doing this. Hardly sufficient.
You might have better luck on some of the islands. Fingers-crossed, it'd be a great place to explore by SUP!
Tim, if you hear back from those you've reached out to, please post. I also reached out to kayak tour companies. We will be on the Amalfi Coast in mid-September. I have seen online that SUP is popular near Ostia, but I doubt we'll make it up that way.
I've been looking into this as well. I sent out an email to a scuba diving and boat charter company to see if they had any insight. I haven't heard back yet. We are planning on being there at the end of September.