How do you politely tell them that they have been denied because you can't get a background report?

If debt was opened but hasn't been reported on the credit report and you deny that any debt was opened, that's fraud. 

We went through this several years ago when we bought my aging mother an expensive condo with 40% down.The banks were not that friendly as it was NOT going to be our principal residence (but we held title for obvious reasons).So we sat down with an independent mtg broker only to get a mtg from one of the banks that originally denied us???

I went for a variance and the neighbors protested and  (NIMBY) I was denied the variance. 

The bank denied me because of Debt to income Ratio.

"...but something about ordering $10k worth of hardwood flooring off a website has me freaked out..."     ..better to think twice, than ask for a refund (and getting denied) once.That being said, I'm curious to know what the 'experts' here think also!!

I would not just present them a letter in person they can deny receiving it.

If at some point you get sued, you want to have meticulous records or why you accepted or denied applicants.3.

Boom, loan denied, invoke loan contingency, get your EMD back. 

They walked away with no injuries or damage to their vehicles, then deny fault/sue me.