It's not unethical if you don't have an exclusive arrangement.
But it is impractical, and I'd even argue it's unprofessional.
Real estate agents are not some interchangeable commodity (like any professional, each has their own specialization, background, education, experience, etc) and treating them like one is not going to win you any popularity contests (or get you a sneak peak at their best deals). If you are "picking out the ones [listings] you like" from Zillow, Realtor.com, or another public site, you are already a day late in many cases.
An approach much more likely to end in success is to actually put in the leg work to meet with and interview a few agents, share your goals, objectives, experience, and expectations, and figure out who would a great agent for you to work with the for the long term. Then treat them like the invaluable team member they are.
Note: I do believe you need a different agent for different markets, or different specialties (multifamily vs industrial or office, for example), but trying to pit multiple agents in the same market (who are looking at the same inventory) against each other is just asking for trouble, with the potential for commission disputes, ethics complaints, etc.
Finally, note also that realtors have a code of ethics that, among other things, prevents them from working with another realtor's clients. That's why one of the first questions we often ask is "are you working with another agent?"
By the way, what does any of this have to do with assumable VA loans?