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Beat cream cheese and butter in large bowl with electric mixer on medium speed until light and fluffy. Add confectioners’ sugar and vanilla; beat until well blended. Stir in chocolate chips. Refrigerate 30 minutes.
Meanwhile, place coconut in large resealable plastic bag. Add food color to coconut in bag; shake until coconut is evenly tinted. Place coconut in pie plate.
Scoop cream cheese mixture into 8 (1-inch) balls with small cookie scoop or melon baller. Place in pie plate with coconut. Keep remaining cream cheese mixture refrigerated. Toss balls in coconut to cover. Reshape into balls if needed. Place in candy paper cups. Repeat with remaining cream cheese mixture and coconut.
Refrigerate at least 4 hours or overnight until firm. Store snow balls in airtight container in refrigerator up to 1 week.