How can pilots obtain official runway length information for airports that do not appear in the chart supplement such as private airports?
If an airport does not appear in the Chart Supplement, the only official information about it seems to be in the relevant Sectional Chart.
In the case of Horseshoe Acres (FA24), the sectional shows that the longest (and only) hard-surface runway is "18", which means 1770 to 1869 feet.
In the case of Brady Ranch (80FD), the sectional shows that the longest (and only) hard-surface runway is "46", which means 4570 to 4669 feet.
Note that you can only guess the runway numbers, the lengths of shorter runways, and the presence of unpaved runways. As @RonBeyer comments, since you have to get permission to land on these fields anyway, it's probably best to ask for such details at the same time. If you get data from the airport operators themselves, that would seem to qualify as official.
However, Airnav and others have lots of consistent data about many airports not in the Chart Supplement, particularly private ones, so it's possible there is some other common source. Since I don't know what that might be, I can't guess how official it is.