if you don’t know call an electrician.

first i would turn off that circuit. you can figure out which circuit that receptacle is on by buying a plug tester at a hardware store (lowe’s, home depot…). Next, I would open the cover to the outlet and see how many wires are entering the box.
If there is only one, you can safely move it in the direction it is coming from and put it in a junction box. preferably on the inside of your house. you can get a blank plate or cover for these in just about any color you want to match the paint on the wall. i would never recommened leaving live wires hanging loosely inside of a wall. that is unsafe, and violates electrical code, and anyone who would suggest that is very irresponsible.
If there are two wires in the box, I would suggest relocating your door, or hiring a qualified electrician to take care of it.
Another thought, I usually run my wires about 30 inches off the floor when going through walls. You might be able to clear the wire…. How big is your dog??

outlet right next to

is there a switch just above or an outlet inside that is close to the outside one

if so,and the outlet is a gfci receptacle ,then there is a good chance the wire is in these boxes
if it is a regular receptacle
then there is a good chance it comes from the other side of the room

just have look and see
either way the outlet more than likely can be moved up the wall

what ever you do DONT CUT ,,TAPE AND STUFF IN THE WALL–
THIS IS STUPID,AND DANGEROUS , AND THE FOOLS THAT TELL YOU THAT NEED TO STOP ANSWERING QUESTIONS THAT THEY DONT KNOW **** ABOUT

You will have to re-route the wiring that is in your way that’s about all you can do if you don’t want that outlet cut the wire and tape the ends up and stuff back in the wall

PS you must check to make sure that outlet is not “feeding” another outlet down line

TURN THE POWER OFF TO THAT CKT FIRST !!!

it would be a whole lot easier to move the doggy door. If you no longer want the outdoor recepticle, you can either remove the wire from the end that feeds it( probably a light switch next to the door or a recepticle on the inside wall) or you can remove the wire from the outside recepticle, cut the wires back to the insulation, put wire nuts on the wires, wrap the whole thing with electrical tape and just leave it inside the wall.

now i might be the brighest bulb on the tree, but if the doggy door is going where the outlet is on the outsdie, and only one wire feeding that outlet….can’t you disconnect the wire remove the outlet box, cut the dog opening , go on the outsdie and put a new hole for the receipt box and run the wire into it…then install the doggie door ??? i dunno seems eay to me

I will tell you from experience, that when you wire a house, you dont leave that much slack, just enough to have some extra in case something burns up. It is too expensive to leave enough slack to move wires to put in a doggy door. Besides, you probably have holes drilled in your studs and wires ran through them. And even if you did have the slack, you can not move it from inside one stud to another. I know that you dont want to operate on your studs, but I seriously doubt that you will be able to move wires around.