Trade/weapon/armor shop empty

You have to wait for them to restock. New stores will get inventory after some time. I had the same issue but after fast traveling and doing some missions my store are now all stocked.

After extensive testing, I have detemined the inventory is of the shop itself, not the settler, and is stocked immediately when the store is created.

However, one of the more questionable design choices was the decision to have both a Trade and Barter dialogue option on the same menu for settlement vendors.

Trade is the left option, which allows you to access the settlers inventory to equip them.

Barter is the bottom option, which allows you to purchase from them.

Doctors are notable exceptions, the bottom dialogue option allows you to receive medical treatment, while the left one is Barter.