Is it possible to modify the ArcGIS Online for Organization's URL?

Yes, it is possible by modifying the short name of the organization.

Note:
Before changing your organization short name, consider the following:
The short name uniquely defines the URL to your organization; therefore, it is recommended that you carefully consider the name you want to use. There are also important considerations for changing the short name if required.

To change your organization short name, follow the steps below:

1. In ArcGIS Online, click the Organization tab > Settings.
2. In General, navigate to Organization defaults and edit the name in the Short name section. Refer to ArcGIS Online: Organization defaults for more information.

3. Click Save.