Would a Kantian think that having unpaid internships as part of students’ courses of study is morally permissible? Why or why not?

One could make arguments based on a Kantian deontological system of ethics either for or against internships, but the strongest are probably against internships due to their narrowly vocational nature. The main issue you will need to address for writing your paper about this is that Kant's system of ethics is rule-based rather than consequentialist. In other words, Kant would not, as this excerpt does, look at the outcomes of internships, but rather look at...

One could make arguments based on a Kantian deontological system of ethics either for or against internships, but the strongest are probably against internships due to their narrowly vocational nature. The main issue you will need to address for writing your paper about this is that Kant's system of ethics is rule-based rather than consequentialist. In other words, Kant would not, as this excerpt does, look at the outcomes of internships, but rather look at whether their existence was compatible with an ethical system based on rules and duty. 

Kant would begin by looking at the ultimate end of internships within the context of education. A good starting point would be to look at Kant's book on education. Kant believes that the central problem of education is how to develop moral character in a child without constraining moral freedom. Equally important is developing habits of discipline and especially self-discipline. Students, for Kant, should be educated with a view to the progress of humanity in the future, not for simply immediate or proximate goals. Rather than pandering to the desires of students and parents for narrow job training or easy and palatable edutainment, educators, according to Kant, should consider what is best for society as a whole. Thus for Kant, the sort of vocational training implied in internships (of any sort, paid or unpaid) distracts from the main purpose of education as moral and cultural formation to act as narrow vocational training; this would would be problematic. 

You could examine this in light of his notion of universalizing one's moral maxims. If everyone simply trained for immediate vocational goals, our civilization would have no people capable of complex political and ethical thought, and would lack artists, scientists, social activists, and others whose education needs to be broader and more global rather than narrower and more focused on specific jobs (which may well become obsolete in a few years due to technological change anyway). Education should be done by " enlightened experts" rather than outsourced to for-profit interests according to Kant.