Is it realistic that I'll have the required project management experience as a Sergeant to get my PMP? If not, what are my options?

It is. Look at the prerequisites listed at http://www.pmi.org. Depending on education level, it will determine how much experience you need to articulate and account for. That's your starting point. I can send a template so you can build your experience timeline and account for it appropriately if you think you're on the right track to take the test now. You'll also need 40 contact hours before you can take the test. You can get those from seminars and courses that prepare you for your test. That's your next two steps. Then submit your application and see what happens. Once you've been approved, you have a year to study for the test. I recommend a boot camp and then taking the test.

Once you've met the prerequisites of course.

If you're not close on the PMP pre-req's, go for the CAPM if you can. It'll get you in tune with the PMBOK and get's you on the path for PMP. You won't find it as a job requirement because folks don't know about it much. I do recommend it for project coordinators who don't have the experience/education yet but are very capable. It builds the resume and experience, making them more valuable to the team. In a company that is looking to grow its own, they will start feeding you more responsibility so you can get to that PMP cert.

I hope that helps.

To qualify to take the PMP Exam you must have the requisite number of project management hours before you are qualified to take the exam depending on your education level. Most things we do in the military can be equated to the 5 phases of Project Management. You probably have more hours than you think. There is a spreadsheet that helps you calculate it since more recent experience is weighted more than older experience.

Look at the definition of a project in the PMBOK guide, tons of things will count.