I would like to display our photos posted on Instagram on our commercial web site using Instagram API.
Officially Instagram says
You do not need to submit for review for this use case. If you are a developer and you want to display Instagram content on your website, then you do not need to submit your app for review. By using a client in sandbox mode, you can still access the last 20 media of any sandbox user that grants you permission.
So actually no need to go Live mode.
P.S. Here is the link to all cases.