Shortly after leaving the gravity well, continue to the walkway and kill more xenos along the way. Once you've reached the next door, enter the hallway and check the room to the right to find the legendary weapon in this mission and a full set of armor.
No Hope in Hadley's
After killing the lurkers in the maintenance tunnels, follow Oneal as he opens the door. Follow the path until you reach the next room. The mission's legendary weapon (Gorman's Pistol) is found on the table of this room. It's really hard to miss.
Hudson's Pulse Rifle
After escaping the pursuing queen by the taking the elevator, you'll be able to regroup with Oneal. Get your gear back from the bag and exit through the next door. Before following Oneal, check the small room ahead to find this mission's legendary weapon: Hudson's Pulse Rifle
Vasquez's Smart Gun
After getting the personnel manifest, wait for Bella to open the other door. Continue outside and take out the PMCs holding position there. Go through the next door and go downstairs. Open the next door and continue ahead. You'll find Vasquez's Smart Gun in the left corner of the room, beside the lift. This weapon doesn't stay in your inventory.
Hope in Hadley's
After holding off the base entrance, head inside when the gate opens then go left. Check the small room to the left to find a full suit of armor and the legendary weapon, Vasquez's Pistol.
Once you've reached the third AA gun, head to the right to find a container van containing a lot of weapons. You'll also find the game's last legendary weapon there. (Frost's Flamethrower) You can equip the RPG and use it to easily eliminate the guards in the area. Take out the AA gun afterward.
There are six legendary weapons in the game but only four of them can be added to your permanent inventory. The ammo for Hick's Shotgun and Hudson's Pulse Rifle is counted separately so you have to pick them up along the way as well.