Is an unbalanced categorical independent variable a problem for binomial logistic regression? My dependent variable is a yes/no measure; I'm trying to predict the probability of an event to lead to a yes or no. The independent variable is a categorical predictor with 3 levels.
My data is unbalanced; I have an unequal number of events in each of the 3 levels of the independent variable (i.e. 311, 180, 238). Is that okay for a binomial regression or do I need to have an equal number of events across the 3 levels? (I'm not asking about unequal yes/no answers within each level, but about the total number of events in each level). I would appreciate any advice on this.

That is completely fine. Logistic regression will happily work with your data.
Roughly speaking, your parameter estimates will be somewhat less precise than if you had more data in your minority conditions, but you should definitely have enough data for estimation precision to be a minor concern.
