I decided I need a community garden plot this year, and since I do a cardboard cover barrier will plant roots be able to survive under the cardboard, or are they going to suffocate due to a lack of oxygen?
You can make few holes on it, or just a large hole near the plant.
Professionally, a bad, but common, method to do mulching, is to use plastic sheets. Air enter and exit from any hole, so you make sure there is some place to let air to enter, and that the sheet (cardboard) has some separation from soil. Plants will not breath like human, so no need to have large exchange rate. Changes in temperatures (day/night) should be sufficient.
cardboard is as permeable to air as soil is, no harm there, you could inadvertently create a dry spot though... if you have a way for water to get down you will be fine, new cardboard is surprisingly waterproof.