It's actually the build icon you need to tap, its the icon with the saw and hammer on the bottom right of your screen, if you have no icons on your screen just tap your screen anywhere and they should appear.
When you tap the build icon it brings up another set of icon boxes, you need tap on the one with the open cardboard box on it and this opens your inventory
If no icon buttons appear on your game screen at all, I would suggest you delete your game and re-download it from the App Store. Opening the game the 1st time after downloading will automatically download a further 26 updates. Your game should then be fully upto date.
Hope this helps.