Sharepoint - No option to create a new Managed Path inside Office 365 SharePoint online 2013

In short: you can't.
By default only sites and teams are available and that's it. You can't add new one.
Checkout related question here

It is not feasible. Every site should be created under https://contoso.sharepoint.com/sites or https://contoso.sharepoint.com/teams.

Refer to: new-sposite. For your convenience, check the following screenshot.

Read the answer here

Also read this