It is! There is a feature to help creating new organizations that is not currently enabled by default so the first thing you'll have to do is send a note to firstname.lastname@example.org asking them to enable new account creation for your organization.
Once this feature has been enabled you'll be able to access this organization creation page to define new orgs:
Please note that your new organization name must be 32 characters or fewer.
Admin users can access this organization creation page from the menu that appears when hovering over their avatar in the lower lefthand corner, and then clicking on "Create New Organization":
After creating your organizations, you'll be able to switch between any Datadog accounts you're associated with from this same menu.
Once the new organization has been set up, you can invite people to your new org from the Team page the same as you would your regular account.