Help with Grafana Alert for Counting Logs from 00:00 to Current time

I think it’s possible. Of course it is not straightforward and you can use now/log time (.ts) formatted to day only and select only logs, which have logday=today.

Idea (not copy&paste solution) - fix any functional/syntax issues:

count(
    rate(
        {<selector>} 
        | label_format today=`{{ date "2006-01-02" now }}` 
        | label_format logday=`{{ date "2006-01-02" .ts }}` 
        | label_format match=`{{ eq .today .logday }}` 
        | match = `true` [$__auto]
    )
)