Experimental and developmental psychologist here. I can’t address the history of peekaboo but I can address the history of object permanence research. This begin primarily with Piaget where he claimed that because 9 month olds will not search for a hidden toy, they do not understand object permanence. Rene Baillargeon and others pretty convincingly demonstrated in the 1980s that the failure to search for a hidden toy was really a failure of problem solving ability, as they extended it to the A-not-B error. Baillargeon also demonstrated understanding of object permanence in much younger infants, as young as 5 months. This also led to a debate where some researchers (like Elizabeth Spelke) believe that object permanence understanding is innate, whereas Baillargeon argues for early acquisition, earlier than Piaget believed. But here the point would be that a 6 or 7 month old baby being entertaining by peekaboo is not about lack of object permanence.
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.704.7776