When training an image classifier, is it best practice to equally distribute the number of images in each category?
Depending on the method that you use, this answer might be different.

First, you split the data between training, validation, and test splits. Forget about the test set. Keep the validation data in its natural distribution of classes.

You have several choices with the training set data -


Balance the classes using down-sampling or up-sampling techniques
Keep imbalanced classes but attach weights to errors in each class (See som references here: 
https://stats.stackexchange.com/questions/207977/balancing-classes-for-neural-network-training)