Theft, including shoplifting, is a crime of moral turpitude and can be a basis for deportation of an alien. 8 U.S.C. § 1182(a)(2)(i) if the alien is convicted of the offense or admits to having committed the offense. Even if the alien might be deportable, there is an exception if the alien only committed one offense that was a misdemeanor offense and for which the alien was not sentenced to more than 6 months imprisonment.
You were not convicted of the offense and if you do not admit to having committed it, you should not be deportable. Even if you do admit to the offense, you may still qualify under the one-time misdemeanor offense exception I mentioned before.
Now, here’s the problem. The USCIS Form N-400 is the form to request naturalization. It’s a very long and detailed form. You will see that you are asked in question 22 the following: “Have you ever committed, assisted in committing, or attempted to commit, a crime or offense for which you were not arrested?” Based on what you said here, it would appear that the honest answer is yes, you apparently committed a shoplifting crime. If you admit to that, you may trigger the bar to admission for crimes of moral turpitude I discusse above, though if you meet the one time misdemeanor exception that may end up not hurting you. If you lie and commit perjury on the application, that too is a crime of moral turpitude and you may end up deported over that if the USCIS caught it because the one time misdemeanor exception would no longer help you.
You ought not panic yet. I think it is possible to navigate this properly to avoid having this cause you to be rejected for naturalization and possibly deported. But you really ought to have an experienced immigration attorney advise you how to address this issue on your Form N-400 and any USCIS interviews to avoid screwing this up. It will cost you (or your parents) some money in legal fees, but it is likely money well spent. If you need your parents’ help for this, then you’ll need to tell them what’s happened. It is critical, though, that you not engage in any additional crime. Felony crimes, crimes of moral turpitude, drug offenses, etc., are all things that can lead to denial of your application and possible deportation. You’ll want to review your entire application with the lawyer and clean up any other possible problems that might, exist, too, like any federal tax problems, etc. You say you have about a year beofre the permanent resident card expires, so you need to start thinking now about that application and make sure everything is in order. It will go much more smoothly if you prepare in advance.

