Yes, you could pull these weeds by hand and try to not disturb the veggies your planted. Lots of work for sure, can be fun and rewarding. What happened? Did you go on vacation or something? Grins. I am being very serious. I am the laziest gardener in the world. I only do just enough. This will be a bit of work this time but you should never have to work this hard again for this bed!
First I would get some big black plastic pots or lay a tarp down in the shade then, using a shovel, dig up your plants. Try to get the entire little plant and as much roots and soil as possible and gently plop into a plastic pot. Or, if you have a tarp, gently place soil and your plants on the tarp, keep in the shade or cover with damp newspaper. Don't worry about the weeds that come with your plants right now.
Have you ever used a line trimmer, weed wacker? You can rent one and wack all the weeds completely down to the ground. Better to do it yourself or risk using your home owner's insurance on an uninsured friend? Wear safety glasses I have to stress this. Careful near the building. Don't weed wack the building. At this point, could you take a picture of the building and the soil? Really need to see what is happening between the building and the soil. Top priority.
I would then take a rake, rake up the debris and dump on the debris tarp to be taken to the compost pile. Then I would get a round nosed shovel, and double dig that bed. Here is how you do this:
Dig down a good foot at one end of the bed dumping the soil into a wheel barrow placed within one step and turned the direction you want to push it when full. When you see a glob of roots reach down and shake the soil from the roots then throw those roots onto the debris tarp. Later just drag to your compost pile and dump.
Just get a shovel full of soil, bring it up and then turn the soil over, the upside down. Dig a foot lower than the surface of the soil. You'll be exposing those timbers used for edging as well as the building/soil connection.
How do these timbers used for edging look after peeling the soil and weeds away? What insects do you see? Take a picture. Please. Turn to take a picture of the building and soil interface after pulling back that soil. Thank you.
Please cover the soil in the wheel barrow with a tarp if it looks like rain and send us the pictures. Hope to assure you of no other problems otherwise we can help detect any problems that need fixing first, cross our fingers! And save you thousands of bucks.
If there are no problems found at this point you should be looking at an empty hole at the end of your bed...at least 3 feet into your bed. Continuing on down the bed double digging, you now will remove the top 4 to 6 inches of the soil from the next 3 foot section. That 'top' soil contains a large amount of weed seeds so dump that soil at the bottom of the hole you just dug earlier. Great soil but lots of weed seeds.
Cover this top soil with the lower level of soil from this same section. Soil below will have few if any weed seeds and will over time be improved using decomposed compost right on top of the soil. This decomposed compost will inhibit any weed seeds from germinating in the soil beneath the compost. The decomposed compost 'feeds' the macro and micro organisms in the soil and makes beautiful tilt. Not to be confused with fertilizer.
As you dig, your bed will easily be 2 to 3 feet high with soil. Don't worry, keep going. When you get to the end of your bed dump the soil in the wheel barrow into the hole...try to get the top soil at the bottom of this last hole and then covered up with the deeper, more weed free soil.
Rake smooth and flatten using a rock rake (versus a soft leaf rake). Don't be afraid to flip the rake over and use the backside. Then, using a chunk of plywood, throw it on top of the bed and then YOU get on top of the plywood and do a little 'River Dance'...I am not kidding!! By now I am always laughing at myself. Your bed might still seem a bit high but it will compact more in a short amount of time.
Next, I would replant the plants you dug up and kept in the shade and moistened into your newly made bed. I would also purchase fresh starts from Lowes or Home Depot or your favorite nursery. Don't plant in lines; plant the proper distance from the other plants but in sections that cover the entire bed. Lettuce and salad stuff I'd do by seed, as well as peas if you want. Plenty of time left in the season. (Put all kinds of lettuce seeds, carrots, radishes mixed together in a pepper flake shaker and shake onto an entire section. sprinkle soil over the seeds no deeper than 1/8 inch). Grab the plywood throw on top of this section and do a littler river dance. Not an entire number...
Peas, I'd do a section of peas with a trellis for climbing or plunk an old ladder in the middle of the seeds you plant (scatter then push each seed down into the soil a good 1 1/4 inches moving some seeds to fill in holes so all seeds are 3" apart give or take and inch or two).
For the starts, your baby plants and the ones purchased at a nursery, I would use a bit of 5-5-5 Dr. Earth's All Purpose. Half the amounts stated in the directions. Water well. After watering I would coat the top of the soil with an inch or two of decomposed compost around the baby plants. If no decomposed compost don't worry about it. Don't use bark chips, please. Wait until the fall to cover with decomposed mulch 2" thick to prevent weeds or, use a cover crop over the winter. AT this point it is easy to maintain weeds. Get them when they are babies, never ever let weeds go to seed and simply cutting off the top growth will starve them out over time (weed wacker is superb at this when you are more skilled wear glasses)! Weeds are my last worry in the landscape.
Just don't go on vacation after planting without a great hired hand pulling weeds? Grins.
Don't put any compost over newly planted seeds. Once they germinate and get their second set of leaves then add a little of the 5-5-5 fertilizer (let me know what you are able to find, even numbers are great for the leafy greens and baby plants. Otherwise, if you want tomatoes we need to use another fertilizer where the first number of the big triad NPK is LOWER that the P and the K), once flowers begin to show...I am using a 2-5-4 with micro chemistry and bacteria, fungi. Or just use Osmocote 14-14-14 for the entire bed at half the recommended amount then forget worrying about fertilizer for 4 to 6 months. Might not get a huge harvest.
Water shallowly at first for the seeds and newbie starts. Grow vertically whenever possible. Don't water on a schedule, just visit your beds once per day, stick your fingers into the soil to test the moisture. Water whatever seeds or plants need water that day, not the entire bed unless you are certain they need water. Water deeply for the starts and then allow to dry before watering again. Keep the seed bed moist only 1/2 inch down until germination then water deeply and allow to dry out.
Concerning the use of herbicide which is very plausible, you could use Roundup or glyphosate right now to kill all of those weeds. Before spraying dig up your plants and store in cool shade, otherwise all will die. Do it right now while the weeds are vigorously growing, don't chop down at all. Do the spraying when there is no wind or any over spray or drift will harm other plants. You willthen have to wait almost a month before prepping your bed to start anew.
What is that structure behind this bed? What is between the bed, the soil and that building? Is that a potting shed, chicken coop? Is this your home with that heat pump?
Herbicide really isn't important. Double digging will take an hour maybe two. Checking on the foundation of that building might save you thousands of bucks.