What you are talking about doing is technically considered corporate espionage without explicit permission from the client.
Facts (as you have stated them):
- You are performing academic research consuming public data of Corporation A
- You will be taking on an internship with Corp A and have access to Corp A's private information
- Your capacity within Corp A is disjoint from your academic research involving Corp A.
- You are at least entertaining the idea of using Corp A's private data for research purposes without their consent.
So my recommendation is entirely dependent upon the nature of the relationship between your internship with Corporation A and their knowledge of your research project involving their data.
If they knew about your research project before granting you an internship, I see it as entirely appropriate to ask them for permission.
If they did not know about your research project before granting you an internship, you put your internship at risk by asking for permission to use their private data, depending on the sensitivity of the data you would have access to and the personalities of the business members in control of your relationship with the corporation. The risks associated with you violating their denial of your request could be more trouble than you are worth.
If they did not know about your research project before granting you an internship and you publish private corporate data without explicit consent, they have a legitimate claim that you sought your internship with them for the express purpose of gaining dubious access to their private information and you would be in for a world of hurt both legally and professionally. Depending on the fallout of that, you could reasonably expect to find yourself to never be employable again in your field and you could reasonably expect to find yourself in prison depending on the nature of the data that you expose and the political clout that is held by the people that you piss off.
Edit To Add
You may already be between Scylla and Charybdis on this. Look at the potential optics: you have a research project (the nature of which you have not disclosed.) You also have an internship with access to corporate private data of one of the subjects of your research study. Assuming that you do not announce your research project and you take the internship, your standing both within your academic community and within the corporate world will definitely hinge on the nature and results of your research project.
If you take an internship with them and then crucify them in your research project, their quid pro quo will be reciprocal regardless of whether or not they consent to you publishing their private data. With access to their private data, your research methodology will be called into question (perhaps legally) and any potential mistake you may have made will be exacerbated. If your results are flattering to Corporation A, then your academic peers may consider you to be politically beholden to Corporation A.
You may already be in a no win scenario other than to cancel your research project or cancel your internship. You really are not supposed to do what you are thinking about doing.