The Housecarl’s Axe is located in a chest in the North of Utbech camp. This is an enemy camp and will be populated with lots of hostile soldiers. You can choose to take out this camp alone, or you can approach it via longship and raid it!
The armor is found in a chest within a hut at the top of Ravensburg fort. You will go to Ravensburg during the Storming Ravensburg main quest.
The breeches are found within a chest in the central hut in Soham Hideout. You will storm Soham Hideout during Path to the Wind-Blue. You could also choose to raid this location, if you need some help from your crew!
The vambraces are hidden within an underwater cave beneath Besuncen Tor. Head behind the waterfall, dive into the pool behind the waterfall and swim down then up into the cave to find the chest.
The shield is found within a chest in a guarded building in the center of Wandrie. Wandrie is heavily populated with enemies, so you’ll need to be prepared for a good fight to reach the shield. Wandrie could also be raided if approached via the river in your longship.