I'm not convinced about the idea of ground cover around vegetables. They do not want competition - that's why we weed between them - and a mulch would be much better for them. As for the rest of the question - if you decide which plants you'd like you can then look them up to see if they are easy from seed. Ground cover isn't really a term I find useful as it applies merely to the function of a plant and says nothing about its botany or growing requirements.

In the case of perennials why not just grow more of them closer together - they don't need specific 'ground cover' - they can do the job themselves.

A living mulch is pretty common but they do often compete with the crop. I've tried clover and rye before.
Clover works very well but your veggies need to be well established first for best results and your soil needs to be as weed free as possible. Clover sprouts quickly direct seeded.

Not sure the brand of straw i have access to but it is gorgeous. Finely chopped, beautiful texture and a pale bright yellow. Pretty much my go-to mulch for herbs and veggies, and others like basil and greens i plant thick enough to not need a mulch but will side dress/mulch with a pro-mix during the grow season.

You could initially use a pro-mix as a mulch, then try direct seeding some clover thickly on the leading edge, maybe a foot around the bed....but just trying to visualize what your garden even looks like....or what veggies you are speaking of. Some need space to produce well, so competition from other plantings between means less production of crop.

Something fast growing and trailing that has one main root structure may work in some cases, like the sweet potato vines or trailing nasturtiums. Re-routing the growth back into the bed if they escape. I've used them in my brussel sprouts/broccoli/cauliflowers bed before to perk up how dull those plants look most of the early summer.

I have a corn area about 20x20 that did well with clover a couple years. Now will be a pumpkin/winter squash bed this year. (covered and not used last year). I've considered a living mulch, but a thick newspaper border covered with a bale of hay works so well and needs less tending and provides a soft surface off the ground for the crop to rest and ripen nice and dry during rainy weather...

Google 'living mulch'. Someone may have good ideas and successful companion planting advice more current... since i first read of it years ago, my ideas and use may be outdated....