Can someone put a name to this game?

Could use more info on the gameplay.

I'd almost think it is Return Fire (PS1), but it was all vehicles there

It wasn't SOCOM 3: U.S. Navy SEALs, was it?.....

.....http://www.gamespot.com/reviews/socom-3-us-navy-seals-review/1900-6135605/

or one of the "Conflict" games? (Desert Storm, Global Terror, Desert Storm 2 Back to Baghdad or Vietnam)

There's Delta Force: Black Hawk Down and Black Hawk Down: Team Sabre.....

Hm, I've only ever played the demo of the game, but maybe you're thinking of Small Soldiers for PS1?

I believe there are some sorts of vehicles in the game, and it has a competitive 2 player mode. I'm not sure about vehicles in 2 player, though, but I think I do remember some level having a turret in the demo.

There was a video game series called Army Men. Maybe it was one of those?

http://en.wikipedia.org/wiki/Army_Men