Grass ID relies on fine details we can't see in the photo, but its overall growth pattern and appearance suggests Nimblewill or Bermudagrass. Both are warm-season grasses (they are in active growth in summer and fully dormant in winter) and perennials. To get rid of them requires either manual removal or careful spot-treatment with systemic herbicide. Since there aren't any organic herbicides that are absorbed and kill roots like some synthetic herbicides do, that leaves the more tedious task of manual removal. Since these weeds can regrow from rooted pieces left behind, this approach will likely take several attempts to fully remove the infestation.

Unfortunately, the only opportunity to smother the grass to help weaken it before removal would require that the bed be empty of vegetable crops, but by the time that happens (assuming you won't have an overwintering cool-season crop or cover crop growing there instead), the weedy grass will be going dormant as well and won't be vulnerable to that technique.

Miri

Smothering the weeds with cardboard won't work well if the bed is still occupied by vegetable plants that can't be covered. Both species of grasses are creeping in growth habit, and they could easily creep out from under a light-blocking cover being used to try to smother them. You can try it if you prefer, but we expect it won't work to eradicate the grasses. The smothering technique needs to block all light (blocking moisture is less critical and definitely not practical or possible in the case of having desirable plant roots in the same area) from the weeds for as long as it takes to kill them. There is no set time frame for this, as it depends on the vigor of the weeds and how much root energy they have stored-up for regrowth when conditions improve (when the cover is removed). It might weaken them for a time until creeping stems reach the edge and can grow new leaves and photosynthesize, but it won't likely kill them or weaken them quickly.

Miri