After high school, if I apply to do 4 years in the military, could I go to an Ivy League college once I'm done with those 4 years?

It isn't impossible, but it is unlikely.  You can apply to any college at any time. Whether or not they admit you depends on what they're looking for in a student. In my experience, most of the top colleges admit few older students as Freshmen, although they welcome them as graduate students later on.

It's unlikely unless your high school grades and test scores are in line with what they accept. It makes more sense to apply to the Ivies right out of high school if you are a highly qualified high school senior. The Ivies have generous endowments and provide good financial aid to students in need (as do many other private universities). Once you have that Ivy degree, then you can join the military as an officer. Another option would be to attend college on an ROTC scholarship and then do your military service after you graduate. Harvard and Cornell both have ROTC programs, as do Duke, UCLA, MIT and a number of other excellent universities. https://www.collegexpress.com/lists/list/rotc-x-3-schools-with-army-navy-and-air-force-rotc/860/

Could? Maybe. Depends on your credentials. Military service might give you a small boost ... or not.

Why don't you just plan on getting a degree while in the military that doesn't cost you anything.

Only if you meet the academic standards of the university and are a better fit than 90% of all the other applicants that get turned away from "Ivies" every year.

As Amaretta noted, if you are an Ivy League-caliber student, you should apply and go now.  You can probably do ROTC and enter the military (if that is your goal) as an officer.  If the issue is money, private universities often have very generous need-based financial aid.  

if you wish. Ivy league schools are pretty snooty and don't hold ex-soldiers in high regard.