Would it be ok to not be specific over where your last job was on a resume?

Yes you can. Resumes are flexible and should be considered as summaries of your most relevant experience, qualifications, and skills. However, there are circumstances when it is not a good idea to leave a job off your resume.

If you don't at least put the name of the store you worked at it will look odd.  When people look over resumes and see things that look odd they either make note to ask specific questions about it, or just assume the worst that you're trying to hide something from them.

If it's a job you were paid to do then yes add it. If you've only been there for a few days / weeks then don't bother using it.

They want to know the store you worked for, the position you held and the tasks you routinely performed while you were on the job. 

In this job market? Sure, hand in a sketchy application. It will go into the recycling bin and save trees.

What you are suggesting is not an unheard of strategy, but a future employer is going to want to know your work history which can be stated upon an application in lieu of omission on a resume.  Some employers will offer interviews without work history knowledge, some won't.