Is there Anyway way to get the in-game pet store without getting the limited edition?

You could probably recreate it, if you are into building. Basically, it just sells stuff for pets, not the pets themselves. Likely, you can convert an empty lot using the drop-menu and from there, build a "pet store" loaded with pet supplies. I haven't tried this, because I do have the Limited Edition. But give it a go and see if it works.

Otherwise, unless you get lucky enough to find a used copy of it, I have no answer for you. Good luck!