Pardon me while I launch into a layman's biology lesson. I have done a lot of research on this issue. Some plants, such as tomatoes, are self-pollinating... meaning, they do everything themselves. Other plants, such as zucchini and most squashes, grow both male and female flowers and require insects (bees) to pollinate by carrying pollen between the two. In zucchini plants, the male flowers appear first and, in an ideal situation, the male pollen will be delivered to the female flowers, and zucchini will result.
Well... sometimes the job does not get done by the bees. You can pollinate the zucchini by hand, so long as you have both types of flowers on your plant. There are several helpful videos online that show you how. I suspected that this was my problem so I went online to research what I needed to do.
However, upon heading outside to get the pollination process going, I realized that my plants have no male flowers at all. The one in the picture, and the ones that I suspect will soon grow from the baby zucchini-looking things, is female. There is no one to pollinate her!
I am going to let this one grow, just to see if anything happens... but, after diagnosing the same problem on our other 5 zucchini plants, I removed them and planted new seeds in their place. I will start from scratch, keeping a closer eye, and pollinate them by hand if needed. There does not seem to be a shortage of bees in our yard, but clearly something is not working (far be it from me to understand why the bees, reputed to be such hard workers, are not pulling their weight in my yard). And I am bent and determined to grow some zucchini! All the resources I checked say you can still plant from seed into July in our region, so I'm hoping something works.
If you are growing zucchini, be sure to take a look at it. If the leaves seem to be growing like there is no tomorrow without producing any vegetables, you may find yourself in this same weird situation (I'm not sure it is really all that common, though). Refer to the links I've included above!