I think those information, health and ammo, definitely have to be available to the player, as they are essential to allow the player to make an informed decision about "pressing the trigger".
Ideally, I would like to have diegetic spatial UI conveying those information, I would like the Vive controllers to be rappresented in game as glowing hands, with the glowing dimming as your amount of bullets diminishes, while for the health, I believe a possible narrative for the game is that you are in a dream world, and the attackers are nightmares, therefore I could have a simple block environment, which gets corrupted as you lose health (textures get a tint of red, smoke particle appear and increase in size).