You can't create a user mailbox in Exchange Online; to do that with the web browser, you'd need to do it in the Office administration portal. But you can do it with PowerShell with the New-Mailbox cmdlet:
$password = Read-Host "Enter password" -AsSecureStringNew-Mailbox -Alias alara -Name alara.rogers -FirstName Alara -LastName Rogers -DisplayName "Alara Rogers" -MicrosoftOnlineServicesID alara.rogers@liquidmercurysolutions.com -Password $password -ResetPasswordOnNextLogon $true
This creates a mailbox for the user, but you'll need to assign a license in the administration center, or the mailbox will be disabled after the grace period.