Nikon lens for sport photography?

I'm looking to get started in sport photography, specifically football, but I'm on a budget. I'm in school and would start out by shooting for my school teams. I have a nikon D5000 (with 18-55mm vr + 55-200mm f4.5-5.6 vr) but these lenses are not long enough for sport and I was wondering what lens is needed for football?
I have found the listed lenses and would like to which ones you think I should get or whether you know of you have any other lenses?
1.sigma 120-400mm f4.5-5.6
2.sigma 150-500mm f4.5-6.3
3.sigma 100-300mm f4 (no os)
4.nikon 300mm f4 (no vr) (450mm on D5000)
5.nikon 70-300mm f4.5-5.6 vr

Of all of them, the 300 mm f/4 would be the best if you are on the field shooting day games.

The other would be the 70-300 mm lens, but will become problematic if you shoot at night.

I use a 300 mm f/2.8 and shoot at 3200 ISO when shooting night games.

Sample:

The key is to be able to shoot at high shutter speeds, 1/800th second or faster

You will need other lenses when shooting basketball or other sports.

I shoot football on a regular basis for a couple of different leagues and high school. I use a 70-200 f/2.8. You NEED the f/2.8 if you are shooting night games/under the lights. The others with their high f/'s will be a PROBLEM when you have to shoot under the lights. The variable f/'s will hurt even more. If anything one of the fixed f/4 lenses will work, but under the lights or inside a gym you are likely to be maxing out your ISO and still being slightly underexposed-which is a recipe for disaster with noise.
The following image was taken at ISO 12800, f/2.8 and a shutter of 1/400 (which is usually too slow for football.) It's rather dark… And had noise removal like mad in Adobe Camera Raw.

That is at a school whose lights are very good. There are many others I shoot at where the lights aren't all that bright.
Your camera can't shoot at 12, 500. You max out at 6400.So… That image would have been half as bright as mine on your camera using the same lens. If you used an f/4 lens it would have been almost black with parts of the white jerseys showing. If you had used the f/5.6 of the zooms you'd probably be seeing only the very brightest whites of those jerseys. With the 6.3 you might not have even seen the blown spots in that image.
Now if your football is all daytime? Any of those lenses will work. If you are shooting inside a gym? Those super long lenses will hurt. I find myself often shooting with my 24-70 f/2.8 on one camera and the 70-200 on a secondary. I use the 24-70 and RARELY the long lens.

You will be needing to shoot at a shutter speed of about 1/500 to get clean, clear photographs in sports so the OS/VR option is nearly irrelevant. The rule goes that to hand hold a camera the shutter speed must be faster than the reciprocal length of the lens. So a 200 mm lens would require a shutter speed of 1/200 or faster. The VR/OS feature actually should be turned off at that point because the electrical current in it can begin to CAUSE blur issues.So, don't worry so much about the VR/OS feature.

I'm in agreement that 200mm on a crop camera can be fine for football. The middle of the field is a bit of an issue, but let's face it you're not going to get every play.

When I'm shooting at night, I use the 70-200mm F2.8 only. During day games, I also have a 100-400 F4.5-5.6 on my second camera. A tripod can be very handy for football as well.