Should this sentence use the singular "is" or the plural "are"?

Solution 1:

Neither. Use "...children receive..."

The present continuous ("is/are receiving") should be used for recurring events that are not expected to continue into the foreseeable future. If this were the correct usage of the present continuous, it would be plural, since children is a collective noun.

The present simple ("receive") should be used (for want of a better reference) for recurring events that you expect will continue to occur in the foreseeable future.

On any given day an average of 400 children receive care in one of seven major programs...

You got it right in the last sentence:

Many children receive multiple services...

I agree with eyeballfrog's second point, that the first sentence needs to be simplified. Maybe try:

On any given day an average of 400 children receive care in one of seven major programs at the Family Healthcare Campus (Child Haven), in our High Risk Infants clinic, and in our Behavioral Health Program. Many children receive receive multiple services[...] Service include home health care, respite services, medical day programming, early intervention therapies, and well and sick health care.