I you start the year on junior status but is supposed to be a senior, can I still graduate that year?

You didn't tell us how many classes per semester of summer school you were supposed to take, which classes you need to retake, nor how many credits you need to recover. Without that, this question can't be answered.

It also depends on what type of program you're in. I'm in a high school program where it is REQUIRED to take four years of math, science, English, and history and three years of foreign language. However, my school district only requires two years of foreign language and two years of math, science, English, and history. However, I will not be able to graduate if I do not have four years of math, science, English, and history because I would not have fulfilled the requirements of the school program I am enrolled in.

You're best bet would be to talk to your counselor about your programming. Your counselor if the best "expert" you can get. See if you can take classes on campus or off campus before school starts. See if there is an after school program for credit recovery. Do anything you can!

And next time, don't slack off.

College/university is MUCH harder than high school. In college/university, they're not as caring as high school teachers. They won't call you after class to ask you what's wrong, to ask you if you want to retake a test, ask you why you didn't turn in your homework. Some couldn't even care less whether or not you show up to class!