Yes, it's possible. With a major credit card (or a check, or money order) you can order a gun on the Internet. I was surprised too. I'm going through it now, so I thought I'd write about it.
Ordering a gun online is not an identical experience to picking out a book on Amazon and having it show up at your doorstep two days later (exception: a Curios and Relics license, to be defined later). An intermediary is required to stand between you and the online dealer. This intermediary is a Federal Firearms License holder, whom I will abbreviate as an FFL. Any brick and mortar gun store or pawn shop that deals in guns will have one of these licenses. So will small, one-man operations, like a guy who does gunsmithing work out of his home - if he wants to receive guns in the mail to work on, he'll have an FFL. The FFL does the transfer - the gun goes from its point of origin to the FFL, who handles a few administrative duties and then sends you on your merry way with your internet gun.
What are these administrative duties? The exact same things that happen when you buy a gun at a shop - filling out an ATF Form 4473 and waiting for the FBI NICS background check to clear. Buying a gun online, therefore, is not a way for a felon or someone adjudicated incompetent to pick up a piece "no questions asked." You still have to hit the streets for that - perhaps the streets of Chicago, or New York City.
Buying a gun online is also not a way for a law-abiding citizen to get around state or local regulations. Bud's Gun Shop, the online retailer where I bought my CZ, is extremely clear that it, for example, will not ship magazines with a capacity of more than 10 rounds to the states that forbid such things (MA, CA, etc.). I have no idea if a smaller operation would do so, but if they did they (and you, the recipient) would be breaking state and quite possibly federal law as well, which I do not recommend.
The FFL will charge for his time - all FFLs charge a transfer fee. The average seems to be $25. I found a guy who does it for $10. I have a feeling that if you went to a gun store they'd charge you a good deal more than $25, since you're taking away their business. I recommend sticking with the home businesses, or maybe a pawn shop.
In many states, face to face transfers (as long as both parties are residents of that state) with no background checks are entirely legal. This is when I, a private citizen, meet face to face with another private citizen in order to strike a deal that involves one party disposing of a piece of property and the other paying for it. In a nation that values private property rights, this strikes me as entirely appropriate and nothing to be worried about or scared of. Some jurisdictions require a record be kept of the transaction, like a bill of sale. I think that even when this is not required, it's a good idea. I would also consider asking the purchaser to produce a CCW or some kind of proof that he's not crazy. This concept is often described as the "gun show loophole," even though from what I have heard most gun shows these days are just filled with retail operations who charge full (or higher) price and who will run all the checks on you. The internet has apparently stolen all the good deals from the gun shows.
Note on Curio and Relic licenses. A C&R license is a particular type of FFL that allows an individual the ability to receive in the mail certain types of firearms, as defined by federal regulations and the ATF. One thing that puts a firearm into the C&R category is its age - if it's 50 years old or more, it's a C&R. If you're interested in WW2 firearms, for example, and a C&R holder, you can buy all kinds of things off of the internet without having to find an FFL to do the transfer - it can be sent right to you. There are restrictions, of course, but considering that guns made in 1962 and before are still popular and useful, a C&R FFL can be a worthy investment.