In SharePoint Online (O365), you can use remote PowerShell to create site collections when you have tenant-admin permissions.

First you need to download and install the SharePoint Online Management Shell.

Run the Management Shell and connect to your tenant-admin url:

Connect-SPOService -Url https://YOURTENANT-admin.sharepoint.com -credential USERNAME@YOURTENANT.onmicrosoft.com

The Management Shell will now ask for your password. After that you can create a site collection using the below command:

New-SPOSite -Url “https://YOURTENANT.sharepoint.com/sites/SITENAME” -Owner “USERNAME@YOURTENANT.onmicrosoft.com” -StorageQuota “500” -Title “My Site created via Remote PowerShell”

Full details of the available parameters can be found on MSDN.