OK, so is this a good place for spare parts?
Yes, it's the best places for computer parts on the web. However, not if a part is an actual OEM proprietary item, they come from the manufacturers.
Machine is just a year old. Could it get dirty that fast? It is in a carpeted, A/C'ed room. Windows never opened, clean house.
This is dependent on usage. The machine does suck all it's air from the room. With that said, a failed fan is also predicated on usage as well. There's no actual shortcut here. You have to open it up, diagnose it, then open it up when you get the proper parts, (if it does indeed need them).
I'm really out of my element here, dealing with laptops. I don't know if their heat sink fan assemblies are proprietary, or perhaps come with the CPUs as a desktop would. Most makers tweak the CPU cooler anyway, and don't just slap on a stock Intel cooler.
When you have this apart, find a metric ruler, and measure the fan, both blade diameter, and bolt hole center to center distance for comparison to aftermarket parts. How many wires does it have, 2 or 3? This is important for determining if you need a PWM speed controlled fan.
At the end of the day, it actually might be best to grab a direct replacement from Gateway, but they'll hammer you on the price, just the nature of the corporate beast.