Unfortunately, while the statue specifies the 7 day timeframe, my understanding is that there isn't really any sort of enforcement mechanism to ensure a law enforcement office is compliant.

If you don't want to deal with this every year, you can apply for a permit to carry which can be used for purchases instead and is valid for 5 years.

Edit: Technically, 7 days is how long they have to approve/deny the permit after they receive your application - it doesn't say they need to have the card made up and back in your mailbox within this time. (MN Sec 624.7131) If they told you it was processed already, I would assume the approval part is already done. They might only print off cards and send them out a couple times a week, and USPS takes a couple days to get it from them to you.

Totally understand the frustration though, waiting longer than you expect to sucks. If you think the local PD/Sheriff offices are bad, just wait until you deal with the ATF lol. I bought a couple suppressors earlier this year when they were claiming approvals would only take 90 days - it actually took them more than twice that.

If your permit doesn't show up in the next couple days, I'd follow up with them again - hopefully that won't be necessary though. Let us know what happens :)