So the dog door is built into an exterior wall in a heated/living space or in a unheated space like a garage? If it's in the garage/unheated space you may not even have insulation or a vapor barrier, it all depends on where you live and if the builder was cheap. Only way to know is when you remove the dog door and can see the insides. Most likely your layers are the siding, vapor barrier of some kind, insulation, then drywall. Repair the hole in that order. Maybe you can find someone at a local home hardware store that can help with what vapor barrier and insulation you need.
If the interior wall is drywall, cut a new piece of drywall to fit the hole as flush as possible, set it aside and in the hole build a box of 2x4s, with the small edge facing you, so that half the edge is under the old drywall and the other half sticks out so you have something to screw your new drywall piece into by using drywall screws. Barely sink the screw heads into both the old and new drywall. Screwing in both old and new drywall into the stud adds support. Use drywall patch to smooth over all crevises and screws, let dry, smooth the patching with sandpaper, then paint.
There are probably plenty of YouTube videos on patching large drywall holes.