Current 2/c cadet here- Government major. My class just went through the process of applying for internships last fall. The process is competitive, and mostly internal within the academy, meaning that you don't apply to the agency directly, but the academy coordinates with each agency, which offers a limited amount of spots for the academy to fill. I was fortunate enough to receive a internship opportunity through the DIA, and I know other cadets who have internships lined up at NGA, FBI, and NSA this summer. Though they did not offer one through the CIA this year, the staff would do everything in their power to try and get you an internship of your interest.
As far as majors go, each department coordinates their own opportunities for internships, so your major somewhat dictates what internships you could apply for. I can only speak to the opportunities offered through the humanities department, but I have seen other cadets go outside of their departments for internship opportunities (i.e. An ORCA major got an internship at DIA through the humanities department). Hope this helps, good luck!