Hi Tennic,
You can acheive this by adding a linked field to the Activity table. This would be AccountNumber field in the Entity table. There already exists a relationship between Activity and Entity, so adding this linked field is possible.
Next, please make a copy of the CRM_NewActivity screen if you haven't done so already, so that you can edit it.
The "Lead / Customer / Vendor / Employee Details (edit...)" area you mention is under the Contact Info section. Finally, drag the new linked field onto this section
Please let us know if you have any questions about this.
-Russell