How to identify if there is a relationship between 5 categorical independent variables to a binary dependent variable?
One simple, tricky but probably best approach would be using a Decision Tree. After building a model, you can analyze the structure of the tree to identify the relation. Afterward, you can list the feature importance to see the effect of an independent variable on the dependent variable.
In python, you can use sklearn to achieve it. feature_importances_ will give you Gini's importance of the features and the plot_tree() function to see the structure of the tree. This article might also help to see how to do that.