Does having a lymphangioma predispose children to getting sick more often?

No. The structural defect of the lymphangioma is basicly an overgrowth of normal lymphatic vessels that form a cystic mass. Although most common in the head and neck, they can occur over the trunk & elsewhere. I find no mention of increased illness in the written literature and have not seen any in those I have followed over several decades.
No. Lymphangiomas are congenital malformations of the lymphatic system. They can get infected but do not predispose a child to frequent illness.