I came across these sentences in the same manga.
[1] よつばはこびがかりな!
[2] しんぶんとってくるかかりか?
[3] よつばちゃん新聞係?
I'm guessing it's 係 "person in charge" in these sentences. Is it がかり whenever it is a verb stem / noun like はこび ? And かかり for everything else, such as verb clauses like しんぶんとってくる that modify it? So I'm guessing the third sentence is 新聞[係]{がかり}.