Having performed my "due diligence...
in an attempt to find a definition of "military ammo", I was defeated. I thought perhaps "ball ammo" would be thus defined, but alas it was not. Ball ammo is only a part of the military's shoulder-mounted small arms munitions inventory. I 'took a shot' at NATO designations, thinking the difference between a cartridge designated 5.56x45NATO (measurably higher pressures, higher velocity, more FPE, and greater effective range) vs the civilian designation of Remington .223 , or the 7.62NATO vs .308 Winchester (essentially the same differences as 5.56 vs. 223) might shed some light on the discussion. Again "fail".
The ball ammo didn't satisfy the definition because our military has access to (and, I would imagine, uses that to which they have access) the "Cartridge, 7.62mm, Frangible, M160" (see US Army TM 43-0001-27). That, and the Hague Convention of 1899 - to which the USA was not a signatory. The position of the USA on the Hague Convention has consistently been: "In 1996, in a legal review of the Fabrique Nationale 5.7x28mm Weapon System, the US Department of the Army stated: The United States is not a party to [the 1899 Hague Declaration concerning Expanding Bullets], but has taken the position that it will adhere to the terms of the convention in armed conflict to the extent that its application is consistent with the object and purpose of article 23e of the [1907 Hague Regulations]." Although this particular example is caliber-specific, the position of the US has not changed over the past 100 years. You can read the position without the part in red - basically it is, "We didn't sign it, and the extent to which we will comply with the 1907 HC is determined by our interpretation of the consistency of application within 'the object and purpose of article 23e' of the 1907 version".
I'm certain somebody will disagree with the validity of my research, and with my opinion that - at the very least - the "military" part of this ammunition discussion is ill-defined. Thankfully IANAL, and therefore I tend to read things in plain English - avoiding idiom, cant and jargon whenever possible. Please feel free to do your own research, and offer your own thoughts and opinion(s)...
As for the legality of "military ammo" for civilian use, I should think "Step 1" would be to successfully define the term itself. Ammunition bearing the NATO military designation is currently available in many places, both in-store and online. I would suggest that if it were not "legal" for civilians to possess, it would not be legal for commercial enterprises to offer "military ammo" for sale. I did not do my research in the hope of starting an argument. I did it simply to try to shed some (what I hope is) generally acceptable light on the OP's topic. Pax...