How are "seedless" fruits and vegetables reproduced?

The companies make them.

Seedless plant varieties can be propagated either by vegetative means like cuttings or cultured from cell snippets or are the result of a hybrid cross between varieties which produce seeds.

Seedless fruits can develop in one of two ways: either the fruit develops without any fertilization (parthenocarpy), or pollination triggers fruit development, but the ovules or embryos abort without producing mature seeds (stenospermocarpy). Seedless banana and watermelon fruits are produced on triploid plants, whose three sets of chromosomes prevent meiosis from taking place and thus the plants cannot produce fertile gametes. Such plants can arise by spontaneous mutation or by hybridization between diploid and tetraploid individuals of the same or different species. Some species, such as pineapple and cucumber, produce seedless fruit if not pollinated, but do produce seeded fruit if pollination occurs.

Lacking seeds, and therefore the capacity to propagate via the fruit, the plants are generally propagated vegetatively from cuttings, by grafting, or in the case of bananas, from "pups" (offsets). In such cases, the resulting plants are genetically identical clones. By contrast, seedless watermelons are grown from seeds. These seeds are produced by crossing diploid and tetraploid lines of watermelon, with the resulting seeds producing sterile triploid plants. Fruit development is triggered by pollination, so these plants must be grown alongside a diploid strain to provide pollen.
ex wikipedia

Fruits are developed around a seed. The seed only occurs when the flower is pollinated. If auxin is added to the flower before pollination the fruit will start to form without the seeds. This is only the case for some plants. Others are done by breading sterile plants, in these cases the seeds come from the parent plants. Vegetables are always seedless.

It depends on the specific plant. But for example bananas are seedless due to one specific mutation some years ago. When that mutation was found it was then asexually reproduced. Either from cuttings, tissue cultures, grafting, etc. This means that they can then make near infinite amounts of banana trees that bear fruit with no seeds. Naturally the tree would not survive and would be naturally selected against due to its inability to successfully reproduce. But humans cultivate them for their uniqueness, much like grapes. They would die if we were not helping them survive. This is why all bananas you eat are clones of every other banana. The genetics that gives them the seedless and desirable quality is preserved and grown by different farmers with different techniques. But they are all genetically the same. Some may taste better but that is due to the different farming techniques used, not the genetic diversity... that was sort of tangent-y.

But the point is that they would not normally reproduce and nearly all of them are found from a mutation that naturally occurred and we kept that specific plant and cloned it a large number of times.