re: "You can go online and buy a gun without a background check"

My knowledge and experience with this all:

Online gun purchases: from several stores (J&G, Whitaker); I purchased, sent funds to the store, they shipped to FFL, I filled out the paperwork and the FFL ran the check.

From Gunbroker, Gunsamerica and Auction Arms (now gunauction, I think): auction bids, won from both private sellers and stores (Coles Distributing Inc, Robertson Trading Post- they use the auctions as storefronts).
Same as above, I sent the funds and received via a local FFL.

I have also bought locally through both pawn shops and Jim's (local gun store). I paid, they ran the check.

In all three above formats, I have had both straight walk-out the door sales, and delays. Never been denied, always have had the FFL check.

Gun shows- same as above. Always ran the background check... but I have also always bought from a vendor at a table.

The possible "loophole" angle they might be talking about: I have responded to a classified (for sale) post in a local forum, made contact with the seller, and met face-to-face to complete the deal. This is legal in Louisiana, and other states too. No paperwork required or done. Guy simply wanted to move one of his Winchester rifles, for a fair price. We met in daylight in a Walmart parking lot, very cordial deal, very nice fella. I paid less than I would buying used from a store, he got more than selling used to a store.
This, frankly, is no different than responding to an ad in the classified section of the newspaper, which is quite legal. We met in public, since that is considered the prudent way to do it between strangers.

Other possible "loophole" I have partaken of: the legal definition of a firearm states that guns older than 1899 are "antiques" and not firearms. This is a federal law, I understand some states place restrictions on it, but Louisiana and most others do not.
I have purchased an Argentine Mauser (model 1891). Mine was made in 1894, has the Ludwig Lowe Berlin stamps on it. Since this is considered an antique, it was listed as such on gunbroker, and required no FFL. Seller asked for proof of age, which I provided. He shipped it to my house. I had a smith check it out and verify it was safe to shoot, bought some ammo, and we've enjoyed shooting it. Nice round, 7.63x53, it's comparable in ballistics to .303 British, and I've seen a few sporterized as hunting guns.

To summarize: the "loopholes" are that it's legal for 2 private parties to sell to each other, as long as you legally can purchase it. You can buy your neighbor's old gun if you want to, as long as you can own a gun (of age, no felonies, etc). The internet is simply replacing the newspaper in classifieds, as it has for every other part of media.
As for antiques, that has been in effect for close to 50 years, and I think it hinges on the change to modern, smokeless powders. For functional shooters, there are not a lot, but a few that make the cut and can still be used today, as long as the gun is in good shape:
Mausers like mine, as long as they were made prior to 1899;
SOME Mosin Nagants are old enough;
SOME Winchester rifles.

I'm sure there are others, but the ammo can be hard to come by. Not so for the above- you can get new 7.63 Argentine ammo made by PPU, you can find 7.62x54R for the Mosin (including new by PPU, as well as surplus), and of course you can find .30 WCF (30-30) everywhere.