How do major online game stores set Magic: the Gathering card prices?
I think is highly unlikely that you'll get an "insider" answer here. Your main question is:

"Is there any public information on how major sites price their cards?"

And an straight forward answer would be: No. Wizards of the Cost does not set prices on ANY card, it all comes down to how much people pay, supply and demand. Whenever a new set releases, I'd say there is some sort of standard pricing depending on the card rarity. This is why, very close to the release, you see most Planeswalkers cost around $20, and then those with a high demand go to around $40-$50 pretty quickly. It is with time and use of the cards, demand or performance of different decks, that prices settle at some point. For a while at least. It is easy to predict, for example, that the fetchlands that are now in standard are going to see their prices go through the roof once Khans rotates out of the format.   

As for automation, TCG has a price aggregator (think on how pricegrabber.com works), and they get information from different sources. They are not actually pricing anything, just showing you lowest prices. Most of this information (think of services that provide them) are public (for a price). So, technically, you can study the behavior on the price of different cards and kind of "predict" prices of similar, future cards, with some margin of error. At the end, the price of a card depends on perception of the value it can have on a given deck. And this is pretty hard to put on code.