There is no copyright in animals (although Patent law can apply to selective breeding and genetic modification, but you aren't doing that). The pet isn't going to sue ;-)

Whether or not you need the owner's permission depends on where the photo is taken. In their home, where there is a reasonable expectation of privacy, you can't take any photos without consent. In public places, or on your own property, I can't see any basis for objection. A pet is the same as a car or other possession, here.

As far as displaying the photo on the web, it's your photo, your right to do what you wish. I hope we are not going to start worrying about internet weirdos grooming pets.