A schwa sound is a short, unstressed vowel sound. It's often represented by the symbol ə. In "family", all the vowels are stressed:
* fam - the "a" is pronounced as in "cat"
* i - the "i" is pronounced as in "bit"
* ly - the "y" is pronounced as a vowel similar to "ee" in "see"
Therefore, there is no schwa sound in "family".