Yes, that PSU will work fine without any problems. It should have all the cables you need and provide enough power.
I want you to consider getting a more expensive, better PSU though; especially if you plan to overclock. That PSU ISN'T 80+ certified, has NO PFC, and won't give you much extra room since it's only 550W.
If you are going to spend that much to upgrade your graphics situation, you need to spend more time making money to upgrade the PSU properly. Don't skimp on the PSU. Stick with the brands DNDHATCHER mentioned and pay a bit more for good quality. It isn't worth upgrading the GPU while skimping the PSU because you increase the risk of killing your entire system. If anything, put priority on a stronger PSU than strong graphics.
Okay, I know it'd probably be a better idea to have a single 4870 or 4890 instead of two 4770s (the 4770s don't perform well for high resolutions because of their limited memory).
But I'm buying this for somebody that's outside the US. He'll be paying for all this and every dollar counts. He plans to use the system for at least 3 years from now and wants the system to be easily upgradable. He wants to buy only a single card now, but if required, also wants to add another 4770.