People have been modifying plants for thousands of years. This is called plant breeding. There are lots of ways to get plants to reproduce without seeds and this was known for centuries before the advent of Genetically Modified Organism (GMO) foods. For fruit trees seedless propagation can be done by budding and grafting. This is how navel oranges (which have no seeds) are reproduced. You take a cutting from a fruiting tree and you graft it onto a different rootstock. The original navel orange tree was discovered in Brazil in 1869. Grafts from that tree were used to create navel groves in North America. Every navel orange tree is still fundamentally a clone of that Brazilian tree. I believe seedless grapes are done in a similar way. Apparently the original seedless grape (which was a naturally occurring mutant) is much older than the original navel orange tree.
You can also pick out the apical meristem, the very growing tip of the plant; you just need a tiny tiny piece. Under a microscope you can tease apart the cells, then these each will grow into a little plant on growing medium in a test tube. These little plants are then planted in successively larger containers until they grow large enough to be trees. This may take a year or two.
There's nothing "GMO" about this. These are breeding techniques and processes that were developed *before* the idea of genetic engineering which is the biochemical creation of specific advantageous mutations in the genome of a plant. Before the advent of GMO farmers and plant breeders had to wait for nature to create a natural mutation that was beneficial. Now we have the science to create any mutation anywhere in the plant's genetic code.
Many plants such as bananas have always been grown from shoots and never from seed.
Many GMO foods *do* certainly produce seeds. For instance GMO rice and corn obviously produce seeds since the corn kernels and rice grains themselves are the seeds. Whether a plant has seeds or not is definitely NOT a way to distinguish GMO foods from natural or organically grown foods. Many natural organic foods are never grown from seed (e.g. tangerines and bananas) and many GMO foods are themselves seeds and grown from seeds.