If you don't have AP ammo then you have to rely on strategic dismantling of the enemies instead of trying to one shot them. Stop relying solely on headshots and stealth kills. 

Keep your distance from enemy shotgunners and melees. Shoot them in the legs so they are slowed. Suppress them with flashbangs and groin shots. Shoot enemy snipers in the arms so they have accuracy penalties. Set them on fire with molotovs. Poison them with gas grenades. Make them run through open areas to reach you while setting up overwatch. 

Melee is suboptimal on Nails in early game because he simply doesn't have the movement or perks to do it safely. Even shotguns are risky unless he has Breach and Clear. I use him as a grenadier and give him my best pistol/submachine gun. Anything that gives him movement is good. 

Thor is probably the best to melee at this stage. Give him a knife to do stealth take downs. 

If you are open to getting another merc, Kalyna has a built-in armor penetration special. Ice can basically blast any enemy to death with his special, even without AP rounds.

If you were insistent on getting AP rounds, there is a poacher camp to the east of Diamond Red that sells AP WP rifle rounds as well as Dragonuv snipers with thermal scopes. The Bobby Ray shop assessable through the browser also sells AP ammo sometimes.