Every time it rains my driveway floods, because it has sunk a little. I was wondering if it is possible to dig under it and block the sunken area up or something or do I have to spend money and get a new one made?

Not knowing the extent of the flooding, or settling of your driveway, it will be hard to say. But NO you will not be able to block the drive up for a resolution. You may be able to bore beneath the drive with a water jet and place a small drainage pipe like four inches in the lowest place and allow the water to pass under the driveway. Of course you will have to maintain the pipe often to keep it clear. And this would only work if you have somewhere for the water to go after it goes under the driveway.