Depictions of girls wearing sunbonnets that hide their faces seem to have begun to appear in the 1870s or 80s, and became a popular icon seen on china, postcards and quilts. It was in 1902 (or possibly 1900, there seems to be some dispute) that the imagery was first incorporated in a popular book series by Eulalie Osgood Grover and Bertha L. Corbett. They nailed down the representation in its now familiar form, though they were neither the creators nor the only marketeers to ply the ‘sunbonnet trade’.
Bertha Corbett created a short-lived Sunbonnet Babies Sunday comic strip series which first appeared on December 8 1907. Her sometimes partner Ms. Grover doesn’t seem to have been involved, or at least wasn’t credited. In the Boston Globe the series ended on October 18 1908. The syndicate responsible for distributing the series is uncredited and unknown, and the strip appeared in papers that took varying syndicate offerings. If I had to guess, I would say the feature was syndicated by a publisher (maybe Rand McNally?) as publicity for the book series.
Alfredo Castelli in Here We Are Again adds credit to Grover on the strip, but I’ve never seen that credit actually included.
Here’s a website with more Sunbonnet Babies history.