For questions where I'm supposed to consult my local orthodox rabbi, if there isn't one, what am I supposed to do? There is a LOR in the city where I live (a Chabad shliach) but he frequently declares himself incompetent to answer questions, since he's not a rav.
You should first decide whether you want to follow Chabad halachic rulings or not. If you do, then ask your local Chabad rabbi who he asks.
If not, then decide what kind of halachic authority you want based on your hashkafa and background. For example, if you feel closest to Modern or Centrist Orthodoxy, then you could contact Modern/Centrist Orthodox Rabbis from the nearest major city. (Or, if there is a major city where you regularly visit, you might look for a rabbi from that city, so you can occasionally attend his shul and see him in person.) You should try calling the rabbi, explaining your situation, and asking if he would mind if you ask him halachic questions when they arise. You might call a few different rabbis to see if there is one who seems the most open to answering your questions, or to see if you have a better rapport with, or closer identification with the hashkafa of, one of the rabbis.
However, you are not necessarily limited to a nearby rabbi. For example, some poskim (such as Rav Shlomo Aviner in Israel) answer halachic questions submitted by email or text-message. See my answer here: Reliable Ask-a-Rabbi Websites
When I have a question that my rabbi can't answer for whatever reason, either he finds me an answer (consults others himself) or I ask him for a recommendation about where to ask. Since you have a local rabbi -- just apparently not one who will answer your questions -- I suggest asking him how you should proceed. He might recommend another Chabad rabbi (as suggested in this answer, or he might send you to someone else. Since he's your local rabbi (that is, the "YL" in CYLOR), he's in the best position to guide you.
Ask the Chabad Rabbi in the nearest town larger than yours. Specifically choose a Chabad rabbi so the answer to your question will be compatible to the things your LOR knows.