Sure, you can plant cool season crops for a fall harvest. But first fill those beds with really good, fertile soil high in organic materials like compost. It's worth it if you must, to buy good compost but check with your County landfill first. Ours here makes wonderful compost and it is free for the fetching. I also make a garden in Utah for my daughter, and their County landfill sells you a pickup truck load for $25, plus they load it for you with a front-end loader, too.
The whole point of raised beds is to be able to grow in better soil than you have in your yard, plus better drainage, and they will warm up sooner in the springtime, usually as well. So getting the best soil for your raised bed is worthwhile. Buying "top soil" is not the best way to go because you really don't know what you're getting - mostly sand usually. Not worth the $$.
Also VERY important - fill the bed, water it and let it settle, then fill again and water. New soil that has been transported compacts a LOT and you don't want your beds to end up half full. Get twice as much soil as you think you will need and mound the beds up plus fill right to the rims.
Next, install micro-sprinklers in your beds. It's fun and easy to do, and like putting together Lego watering stuff. If you get a timer to put on your hose bib, you can set it so the beds will get watered regularly without you having to remember. The amount you water your lawn is not nearly enough for veggies, they need to be watered every day in the mornings right through the summer to give you good results.
Lowe's or HD both give seminars on micro-irrigation. I like the brand of materials that Lowe's sells the best. There are also many online tutorials you can look at. The beauty of the system is it is flexible - you can water the right amount in the right place, and if you have a crop that is finished and you've pulled it, you can turn off the water to one small area. Or water more for seedlings, or way more for big plants in hot weather.
We put this micro system in at our local school garden, 6 large raised beds plus a butterfly garden, for less than $100 including the timers. It took one Saturday morning's work by two ladies. We've since expanded to 10 beds and two timers on one hose runs it all, and keeps everything beautifully watered even over long weekends and vacations.
