I own both and have used both to cover rugby and motor sports. The 70-300vr is a great lens and with a cropped DX body should give you good reach and is easy to use handheld.
The 150-500 Sigma is much heavier and costs more. The OS works well to compensate any photographer wobbles, which will get more pronounced in the second half of the game as your arms get tired handholding! At max zoom length you will most likely need to increase your iso to maintain a good shutter speed in order to freeze the action as the lens is not super fast, but the cost of the lens reflects this of course! : ) I don't know how good the 5100 is at high iso but being a relatively new body should not cause any big problems for you in good daylight, just a point to be aware of.
In summary, I'd say go for the 70-300 vr (lighter, smaller, lower cost, more versitile) and if it is not long enough go for the sigma 150-500 OS, in terms of quality both are good and I am very pleased with the results I get from both wide open. (I'm not a big pixel peeper just to qualify my comment)