Can I become an Army pilot while I'm currently in the Reserves? If so, how soon can I start the process?

You sure can. I'm not entirely sure what the key differences are applying as a reservist as opposed to Active, so I'll let someone else chime in on that. The WO Recruiting website will be a great resource for you while putting your packet together. This link is specific for Guard/Reservist.

https://recruiting.army.mil/ISO/AWOR/Active-Guard-Reserve/

Key things you will probably want to start thinking about is raising your GT score if it's lower than 110 and applying for a secret clearance (clearance would be the longest delay if you don't already have one).

If you are looking at doing ROTC as an option you can do that while being in the reserves. You would sign and commitment to commission as an officer and you would be enrolled both in the ROTC program and in the reserves but you would not be deplorable while you are in ROTC.

The harder part though is picking aviation as a branch. The way they pick these is based off how well you do in ROTC and how well you do at camp which is held during your Jesuit and senior year.

Basically the way it worked when i went through it is the top 10% get their first choice but it is the top 10% after they split every cadet that went to camp that summer into 4 sections. So you can be the 76% overall and still get your first choice.

The more guaranteed route to get aviation is as a warrant.

I would say decide what you would rather do. If you want to fly and fly only then warrant. You do that most of your career until you start making CW4. Vs as an officer you really only fly for maybe your first 3-5 years and they you start getting pulled up to staff.