Calculating the average BMI for children Calculating the BMI of an individual is straight forward:
Divide the weight in kilograms by the square of the height in meters.  

The interpretation of a BMI value for adults (for this purpose the CDC defines an adult as someone age 20 or greater) is also straight forward: 
BMI             Weight Status
Below 18.5      Underweight
18.5 – 24.9     Normal or Healthy Weight
25.0 – 29.9     Overweight
30.0 and Above  Obese

Interpreting the average BMI of a group of adults is also straight forward: average the individual BMI values and use the above chart to interpret the average BMI (i.e., on average the group of adults is Underweight, Normal or Healthy Weight, Overweight, or Obese).
However, the interpretation of the BMI for a child (the CDC defines a child as someone older than 2 years of age but younger than 20 years of age) is a little more complicated.  The BMI value for the child is referenced to a gender specific chart which contains BMI-for-age percentiles for different ages.  The BMI-for-age percentile indicates the interpretation:
BMI                                  Weight Status
BMI < 5th pctile                     Underweight
5th pctile <= BMI < 85th pctile      Normal or Healthy Weight
85th pctile <= BMI < 95th pctile     Overweight
95th pctile <= BMI                   Obese

I would like to calculate an average BMI that I can then interpret as I did for adults, but I am struggling to define how to calculate and interpret the average BMI for a group of children that may include both genders and has children of varying ages.  I cannot take the average BMI value and compare it to a chart because the individual BMI values correspond to different ages and genders.
EDIT (To answer Glen_b's questions and provide more context):
I am looking at a small region (think 3-4 counties worth of area in the United States) of the country divided by Zip Code Tabulation Areas (ZCTAs).  For each ZCTA, I would like to measure the prevalence of overweight and obese people in that ZCTA.  This will provide some evidence as to why one ZCTA is more suited to and will hopefully benefit more from a targeted intervention to reduce obesity than another ZCTA.
 A: *

*Like you did for taking the average BMI for a group of adults.


You first calculate the BMI per individual. After that you take the average of all individual BMI's. And compare the group mean to a chart.
(note that you comparison of average is quite rough. you could have an average BMI of 24.8, yet you know that it would be simplistic to classify the group as normal)
You don't use the average height and weights $\bar{BMI} \neq \frac{\bar{w}}{\bar{l}^2} $ but instead $\bar{BMI} = \bar{\left( \frac{w}{l^2} \right) }$.
A similar logic works for the children:


*

*For the group with mixed ages and gender you do something similar. 


You first calculate the BMI per individual and compare it to the percentiles in a chart (if I am not mistaken these charts exist for >2yrs). After that calculation for each individual you take the mean of the percentiles or use some other measure
(Instead of the mean of percentiles you could use the mean of the relative difference from the mean, z-value, amount of people above/below the median, etc. why? for example because a person in the 50th percentile and a person in the 99th percentile, half the group overweight, is not the same as a two persons in the 75th percentile, the entire population normal. This concept is also a problem in the comparison that you make for adult groups.)
A: Background: If the target population is younger than 18, the convention is to avoid using their BMI. Because children are actively growing, the same BMI from different ages can indicate a different severity of being overweight or obese; so what we needed was actually BMI that is adjusted for sex and age. To work around that, researchers devised this thing call growth references. BMI data from different age and sex were collected from the children, and then based on the distribution in each age segment within each sex a probability curve was drawn. The many probability curves were then combined to return the growth reference charts. There is a set for boys and a set for girls; and there is a set for age 2-20 and a set for age <2. Currently, two commonly reference data are being used: the CDC one and the WHO one. In these growth reference, the unit of BMI (kg/m^2) has been stripped, instead the data are presented in either percentile or its equivalent z-score (e.g. 50th percentile as z-score 0). The take-home message is that these curves were based on reference populations, and were not generated empirically using your own data.
Technical part: Software codes exist for researchers to convert children's BMI into these BMI z-score or BMI percentile. The most commonly used one is the SAS codes provided by CDC. WHO also provides syntax codes for their reference data and codes in the form of SAS, SPSS, Stata, and R. They all have a little bit of learning curve, read the tutorial carefully, make sure to prep the input data set correctly and to check the returned data to see if they are valid on the surface. Notice that neither of there requires you to upload your data to their server, you can do all these processes offline.
Advice: Since you have individual data (about a couple hundred for each ZCTA,) I'd suggest compute all the BMI z-score or percentile for each child using the code above. Then, categorize them into the four levels of weight status. Run a frequency and percentage table for each ZCTA of interest, and then examine which one has the highest prevalence rate of overweight, obesity, and the combination of them. (I'm assuming a reasonably simple random sampling. If not, make sure proper sample weights were applied during the analysis.)
I would strongly advise against using the mean BMI or mean BMI z-score as the benchmark of the whole community, because both are continuous and very much right skewed, technically a community of 500 can be "on average" obese with 499 residents having normal weight and 1 resident being astronomically overweight, or they have Godzilla as their neighbor.
A: Your best bet is probably to calculate a scaling factor for age and gender, either constant or dependent on one/both factors. If it were me I'd transliterate the graph into a matrix with age, gender, and BMI, then take the ratios by gender and age and create a linear fit. Since the two variables probably influence each other, include an interaction term to account for the hetereoskedaciticy.
Then you can go back to your data, adjust all your observations appropriately, and take "blind" averages just like you did with adults.
A: You have to actually download the dataset from CDC to do this. 
www.cdc.gov/growthcharts/cdc_charts.htm
Questions about datasets are off-topic for this site however.
