Some people are getting the coupon upon signing up, others are not. Give it a try.
On mobile I added my first pet ever to Amazon (must be new to the program). After creating the profile I was given a $10 credit. I then located my favorite dog food (16 lb bag of Purina One—tried and alternate food and it worked too) which also had a 40% off subscribe and save. I clicked that and used my credit. I got a huge bag of dog food for free and then cancelled my subscribe and save when I got it shipped.
I could not get the coupon when I tried it via desktop on a relative's account, so try mobile first.