United Airlines currently has plenty of award space for the whole family on direct flights to Europe during summer 2017. Both economy and business class availability is strong from a variety of gateways, with some dates having as many as 8 saver-level business class seats available:
Chicago to Dublin, 4 seats (solid lines indicate economy saver award tickets and dots indicated business class space):
Newark to Paris, 4 seats:
Newark to Paris, 8 (!!!) seats:
Many other direct destinations are available including cities such as Amsterdam and Brussels, with connections onward throughout Europe as well. United charges 57.5k miles one-way in business class for these United-operated flights (versus 70k miles one-way on partner airlines). Remember that round-trip itineraries would qualify for the excursionist perk, meaning that you could add a free flight within Europe to a round trip award.