Really current update beyond the 5-11 numbers discussed in the news this week? Nope, but there's no reported big change from, say, July 22, when President Biden told a town hall in Cincinnati he was told they would be available maybe by the end of August. But as GoofGoof pointed out, there are ongoing trials that should produce results soon. The question then will be how long it will take to get even an Emergency authorization. News reports in the last few days say that Pfizer will submit for FDA approval next month, with expected approval in "late fall, early winter" this year. That's for 5-11, and even that vague timeline is better than for the under 5yo's.
There's no announced timeline for <5yo's that I've seen, and the official word is that it will be really slow. Peter Marks, the head of vaccines for the FDA, said in May that "Just so everyone understands, we are going to be very careful as we get down to smaller children ... We have to reduce the dose of the vaccine, we're more cautious about side effects, it takes longer to do the development." Another hold-up appears to be getting enough test subject volunteers; one reason for the slow 5-11 approvals is an inability to find volunteers in that cohort.