I applied for grocery like 2 days after making an Amazon Seller account. I didn't even have the business ready to go yet. No supplies, no inventory, no experience. I made the account, saw some great sales and bought a bunch of items.
I spent some time reading around and making myself nervous about applying. Funny, I was approved less than 5 minutes after applying for grocery. (I think it was actually 3 minutes, I was shocked.)
I followed the advice given here
Basically:
I submitted the application, walked downstairs to get a drink, and heard the beep telling me I had a new email. It was the approval notice.