I think there is a survey among apprentices that does exactly that. I think Rick Mueller did a study that looked at completions using that survey, which I think also gives apprentices the option to say why they dropped out.
Survey evidence can be informative, but of course the reasons someone gives you may not be the reason they do drop out. People often talk about how periods of employment instability are important. Employers talk quite a bit about math and literacy skills of the apprentices, that they're not up to what's needed in order to get through the program. Yes, there are studies that do exist.