There is no ribbon for outlook web. Your add-in appears in ribbon for outlook desktop, while in outlook web, it shows either as pinning button or as an item in context menu. Please checkout this article for details.
If there is no "Store" button, please try to see if you can find the "Get Add-ins" button which has similar function. If it is still not there, please check if you are in compose mode. The button doesn't show up if you are in compose.
On the desktop version, I see the icons on the ribbon (which is what I want), while in https://outlook.office.com I see it on both the read pane and the compose pane. That doesn't really fit in with the functionality or the add-in. Is there no way to launch it when neither reading nor composing?