I also use the cardboard on clay topped with soil raised bed. It's a good idea to fracture the soil underneath by standing on a garden fork and rocking it back and forth. Also put down something highly nitrogenous like composted manure, rough compost, or alfalfa meal/pellets to help break down the cardboard (especially if you've already removed existing green stuff -- I usually leave grass/weeds under the cardboard) and water the ground and the cardboard well before adding the rest. I wouldn't be planting around here until mid-March at the earliest, but, if you are planting soon and are worried about the cardboard breaking down in time, use 4~5 sheets of newspaper/newsprint/kraftpaper (usually used for packaging) instead, otherwise, your veg roots may hit the cardboard and go into nitrogen deficiency shock.
You know peas (especially if growing snap peas) go in first while it's still kinda cold, right? Then tomatoes after things warm up a bit more (light jacket season). The squash and zukes go in after it's no-jacket warm. You could probably also grow lettuce, arugula, spinach, etc. leafy greens before squash and zukes, and grow bush beans (or the zukes) after peas.