Thanks for the vote of confidence, @Linn. Even though I have been growing flowers and veggies since I was a kid, I'm not sure I can answer this question. There are too many variables. I will toss out some very general suggestions.
Knowing the soil type (texture) has a big impact on watering times. Unless the box is filled with native soil, the USDA soil survey is of no use and a jar test would be more helpful. Since you built a box, my assumption is that it is filled with purchased soil from the local nursery or big box store.
If that is the case, then the soil is most likely a "loamy" type--that is most likely an even mixture of sand, silt, and clay.
As @Linn suggested, take a look at the spreadsheet that @azdavidr put together. For raised bed vegetable gardening watered with a drip system, instead of thinking surface wetting pattern, think in terms of the depth of the wetting pattern. The team at Rachio has put together a great knowledge base with lots of illustrations. http://support.rachio.com/article/262-choosing-soil
Flexible schedules work great for my lawn and even some of my well-established xeriscape, but veggie gardens are unique. Plants at different stages of development may require different amounts of water. Overwatering can cause the same symptoms in the the plants as underwatering, such as drooping leaves, yellowing and leaf drop. Overwatering also leaches nutrients from the soil and prevents oxygen transfer.
Some vegetables may wilt on hot days even if they are properly watered. If they "perk up" in the morning, they are most likely fine. Wilting leaves during dry, hot days are not uncommon in tomatoes, peppers and broad-leaved plants like squash and cucumbers.
The "general rule of thumb" veggie gardeners follow is 1" of water / week. During hot and dry seasons, depending on what is planted and stage of growth, 1 - 2" / week would be appropriate. 1 inch of water applied to 1 square foot is approximately equivalent to .6 gallons.
The best suggestion I can share is "observation". Spend time in the garden (to me that is destress time). Observe plant growth and health. Push your finger into the soil to test the moisture level. With some experience, the soil moisture levels can be estimated quite accurately.