I am looking for a rcipie for a filling I can use in dog cookies.
Something that does not require refrideration, is non dairy and sugar
free. It should look like and have a similar consistancey of the
filling you see in an Oreo cookie.

How about spreading silken tofu on your dog cookies? It resembles Oreo
filling, it is non-dairy and sugar-free, and doggies love it.

Regarding refrigeration:

"Silken tofu is one of my pantry staples. It is very custardlike in
texture and requires no refrigeration."

http://www.nctimes.net/columnists/colarchive/anderson032300.htm

It's amazing how fancy some of the recipes for pets can be, and many
I'd not mind at least tasting, if not eating them. I couldn't find a
recipe that fills your requirements, although I did find a cookie dip.
An interesting site I ran across, though, suggests using doggie
toothpaste as a cookie filling. Apparently the paste comes in several
flavours -- chicken, beef, etc. It's easy to see the benefits of this,
although you wouldn't want to be throwing a cookie to your dog every
hour of the day.
Good luck!

I've seen recipes for non-dairy cheese that might make a tasty filling
for a canine cookie.