I got my FL CCW permit with an embossed notary, not the inked stamp. No problems. Of course I don't understand your "scanned" comment. I did scan my application so I could retain a copy, but I sent the original notarized application to FL.
In my case it took about 6 weeks round trip from MD to FL and back to MD. I just got mine about three weeks ago so I'm guessing that is as current a metric of their turnaround time as you're likely to get, for now.
And to address another FAQ recently, I sent one combined check.
They scan these applications and send them somewhere (don't know where). I called and they said they would like the notary to use a rubber ink stamp because if the embossed stamp did not scan well it could delay processing and I may have to go through and resubmit everything again. Just trying to avoid any hassle and get this done right the first time.
P.S.- Found a notary with a rubber stamp, sent off my app today!