The story is a series of vignettes, each with its own small problem that needs to be solved depending on what is happening in the town that summer. But there are a couple of overarching conflicts that you could expand upon to answer this question.
One conflict is the tension between the life that Joey and Mary Alice live in Chicago and the life they live while they are with Grandma Dowdel. There are a number of allusions to those differences, such as the picture show they attend in town that is put on by the Lion’s Club versus the way they are used to going to a movie in Chicago. These things both surprise and educate the children, as well as cause them frustration.
Another conflict is the inner one that the children wrestle with over their understanding of their Grandma. They are both confused and amused by her antics. At one point in the middle of the book, Mary Alice says, “I don’t think Grandma’s a very good influence on us.” But later, Joey notes that their Grandma has positive traits, saying, “From Grandma, Mary Alice was learning thrift.” Their understanding of her, and of themselves, grows throughout the years and is resolved beautifully in the last chapter.