Which living organisms have antibodies?
Apparently, cartilagenous fishes are the oldest extant clade that have the true immunoglobulins (antibodies) and the T-cell receptor (Schluter et al., 1997). All extant gnathostomes, incuding cartilagenous fishes, produce antibodies.

However other animals (invertebrates) and even plants have proteins that belong to the immunoglobulin superfamily (to which the classical immunoglobulins/antibodies also belong). Members of this superfamily are structurally related; they are not necessarily involved in immune responses. Hemolin is one immunoglobulin superfamily protein in insects, that is involved in immune function, but not exactly like vertebrate antibodies.