Okay, when you buy an item it will appear in your inventory but you won't be carrying it. You will need to go back to the merchant where you bought it from and then click the 'items' button. When you have done that you will see the items and you can select where to equip them to.
Also bear in mind some items haven't been implemented yet. The only items you can use straight away are belts, backpacks, and clothing which grant extra storage capacity. Other items which you buy from factions, such as stun cuffs, you will need faction privileges to use.
Hope this helps.
As a reminder to you, you can use your inventory to let you know which building you bought them from if your goods are still in it.