If you go to a grocery store today to buy grapes, there is a good chance that the only type of grape you can buy is seedless. Given that nearly all grapevines in production today produce seedless grapes, the question about reproduction is obvious.
It turns out that most fruits today do not come from seeds. They come from cuttings instead. This is true of grapes, blueberries, apples, cherries, etc. (pretty much all fruits except citrus). A piece of a vine or branch is cut off, dipped in rooting hormone and then placed in moist dirt so that roots and leaves form. Because they come from cuttings, new grapevines are essentially clones of the vine they were cut from.
Seedless grapes actually do contain seeds at some point. But a genetic error prevents the seeds from forming hard outer coats like normal seeds do. Presumably, several thousand years ago, a grapevine grew from a seed and had this seedless mutation. It has been propagated by cuttings ever since.
This page talks about the technique for capturing seed embryos from a seedless grape to create new varieties.