A caveat: "productivity" is not a term used much by cognitive science, so I'll talk about cognition instead. It's not necessarily the same thing, but it's more of an answered question.
And as a general issue: any time a question has "more" in it, the real question is "more than what?" More productive early in the morning than later in the morning? More productive in the morning than the afternoon? More productive in the morning than adults?
There is little work comparing children's cognition across the daily cycle; a good review of mostly adult work can be found here (Chronobiology International (2000), 17(6), 719–732, doi/abs/10.1081/CBI-100102108). Non-experimentally, there is some research supporting the idea that starting school later in the morning--particularly for adolescents--improves performance.
This blog post details an analysis of test scores from Wake County North Carolina (where start times vary), and shows "delaying school start times by one hour, from roughly 7:30 to 8:30, increases standardized test scores by at least 2 percentile points in math and 1 percentile point in reading. The effect is largest for students with below-average test scores, suggesting that later start times would narrow gaps in student achievement." Obviously, there could be major confounds here: we don't know if start times correlate with performance-relevant but unmeasured characteristics--although the author seems to have done a reasonable job of controlling for measured characteristics, and compares students that are similar at similar schools with different start times.
This New York Times article cites several cases where attendance and performance rose after pushing back start times from 7.30 to 8.40, but again, the lack of a controlled comparison means that there's no particular reason to think that generalizations from this case are valid.
There are problems with this approach, of course--for starters, there's no way to control for the effect sleep duration or quality, which might be expected to covary with start time, on performance.