インターンシップに応募したいんですけど、担当者の方はいらっしゃいますか？って英語でなんて言うの？

If you would like to speak to the person in charge of an internship that you would like to apply for, you can say:
Can I speak to the person in charge of internships? I would like to apply.
'The person in charge' - This is the person who is responsible for something.
I would like to apply - This means that you would like to begin an application for the internship.

I hope that helps!

Human resources would normally be the department in a company that appoints employees.
I suggest checking with Human Resources to determine if the company does have any internships available.

It is critical that you speak to the correct person able to make decisions about internships. In a large company there is usually a designated HR department...or possible the HR function is contracted out to a specialist company and that company would need to be identified and approached. Requesting to speak to people immediately and without an appointment is probably not the best strategy, although it does show a level of determination! Alongside anything you do, another approach could be to write first, requesting an appointment to discuss internships. If there is no response to your mail, you could then follow up with a phone call. In that case the above requests may be appropriate.

This is being very direct in asking to speak to the person that is in charge of interviewing and hiring the interns.

"Who do I speak to about an internship, please?"

Inquiring about...? IE Wanting to know something:-) In this case we need a sentence in order to get through to the right department....Something like this:
"Hi...I am inquiring about internships.... which department deals with that? "

You can say :

"I would like to apply for an internship, may I please speak to the person in charge?"

apply (application)
internship (a student or trainee who works at a company)
person in charge (manager, supervisor etc.)

I am an intern and I would like to speak to the manager please.