Most etymologies I've come across for "ベビーカー" assume that it's wasei eigo based on a combination of "baby" (ベビー) and "car" (the "カー" part of "マイカー"). For example, the Japanese edition of Wikipedia says this in a footnote on its article.
Could it instead be derived from a contraction of the English phrase "Baby carriage"? I suspect that "carriage" is distantly related to "car", for what it's worth.