While Amazon doesn't pay the tax itself, charging its customers more money will make its products less competitive (unless Amazon eats the cost of taxes, which would mean lower profits).
Before 2008, this wasn't an issue for out-of-state Internet retailers like Amazon because a 1992 Supreme Court decision stated that retailers that don't have a physical presence in a state don't have to collect sales taxes for sales to that state.
Starting in 2008, however, New York led the charge for states to redefine what constitutes as in-state and out-of-state. Their main strategy is counting affiliates - or entities that sell Amazon products (e.g. a cooking blog selling Amazon cooking books) - as physical presence.
So if Amazon has an affiliate in located in New York, the whole company would be counted as in-state and therefore be forced to collect sales taxe.
Amazon usually responds to these state laws by killing its affiliate programs in those states. For New York, Amazon kept its affiliate programs but is challenging the law in courts.
For California, Amazon has already threatened to cancel the state's affiliate programs. However, that might not be enough because the law's in-state clause also extends subsidiaries, which Amazon operates in California.
One of them is Lab126, which develops the Amazon Kindle. The Kindle is Amazon's best-selling product of all time. It's also the gateway for consumers to read Amazon e-books, which sold more than Amazon's print books in 2010.
Presumably, Amazon will have to relocate all of its subsidiaries out of California if it wants to avoid collecting sales tax there.
Amazon hasn't indicated whether or not it'll challenge the California law in courts.
Ultimately, Amazon can probably avoid collecting sales tax in California; the only question is how determined it is and how much it's willing to sacrifice.
However, it may be more than just avoiding California's 7.75 percent sales tax.
California is the US state with the biggest economy. Whatever California legislates for products (e.g. safety standards on cars) usually sets the standard because companies can't afford to ignore this market. This process even has its own name: the 'California Effect.'
If Amazon bucks to California, it's unclear whether it'll encourage other states to follow suit. But if it defies California, it will draw a line in the sand that no other state will likely dare to cross.