Question:

How to reorder the Supporting Documents field to put it in the upper section of the Field Layout for all assets in Vault? 

Answer:

The current order is:

1. General Properties 
2. Product information 
3. Piece Details 
4. Sharing Settings 
5. Supporting Documents 

Use the Related Documents section in the Field Layout Admin page to change the order of appearance for Supporting Documents.

The reason the name does not match exactly is that the Field Layout section controls all the sections that list related documents (some of which only appear for a specific Document Type). This includes Supporting Documents, Related Claims, and Related Pieces. The latter two only appear if the Document Type is Promotional Piece. 

Related Documentation:

Vault Field Documentation: Managing Document Field Layouts