We just bought our first house a week ago (literally) and yesterday we discovered (despite nothing being wrong during the home inspection) that our kitchen sink no longer drains. It fills up very fast (~1/2-1 gallon of water run), which made us think the clog was close to the sink. We have taken all the pipes off back to the wall and cleaned them out. We have also tried snaking the line with an 8ft snake, but it hit nothing. We are really confused about how it can be clogging so fast if an 8ft snake didn't hit anything. Could the air vent be clogged? We've had some storms lately, though I hate to think that this will happen every time a storm rolls in.

You just didn't snake far enough, get bigger and longer snake.

our kitchen sink no longer drains.It fills up very fast (~1/2-1 gallon of water run), which made us think the clog was close to the sink. We have taken all the pipes off back to the wall and cleaned them out. We have also tried snaking the line with an 8ft snake, but it hit nothing This tells me that you have a clog in the horizontal line in the wall or it could be that your drain is completely blocked and the line's full from the clog all the way up past the sanitary tee in the horizontal branch. To test this go up to the kitchen roof vent and drop a small pebble down the stack if you hear a splash the clog's farther on down the line and you must snake from the roof vent. Or you could take a hose up there and turn the water on. If the sink begans to fill the clog's in the kitchen vent.
Rent a sewer machine and snake the roof vent. Put out enough snake to reach the base and 20 feet more. Good luck, tom