my brother had a convertible and all too often would leave the top down. Weed grew in the carpet rather well.
I have had some problems with weed blocking ground fabric. Nice for use around shrubs and perennials where you put mulch on top of it, but in a vegetable bed, thing like ants and cutworms, and moles like living under it. Use stuff like pine needles, or grass clippings but keep them from touching the stems of your plants. Personally, I like to just scrap the surface of the soil with a Dutch hoe. Do this often. Remember, every time you till soil, you expose seeds just waiting to sprout. Mulch is really needed around tomatoes so they don't get the splashing of the soil on them which encourages diseases. Don't use mulch until the soil has warmed up. There is a red garden fabric which has shown it does encourage the fruiting on tomatoes. I've never tried ground cover but it might be worth a try.
I have used the cheap weedblock fabric for years and this year bought a carpet remnant 12'x17' for $30. Been wanting to space my tomatoes further than 3' as they tend to crowd each other - most weed block is 3' wide roll. I run the Gilmore fabric soaker hose on my rows using garden staples lay out the covering and then plant. My tomatoes are already budding here, summer squash is ~1' tall, cukes just starting to vine out. A pic from last year:
Well I found a huge amount of weed cloth for $19 at farm and home that came with the pins and all so I bought it. There was enough to do my entire greenhouse floor and 2 garden beds outside which leaves about half the roll. That was 3 layers deep so hopefully it works! The pallet bed ideas all be put off till next year because I have maxed out the garden budget building the greenhouse and buying the plants.