and stay together with the family at the Grand Hotel in LA”.

You could equally say

“Share the good times by staying together with the family at the Grand Hotel in LA”.

In basic terms, this clause tells you how to "Share the good times" - it is a complement of "Share the good times".

No comma is required.

and stay together with the family at the Grand Hotel in LA”.

You could equally say

“Share the good times by staying together with the family at the Grand Hotel in LA”.

To that extent, and in basic terms, the original clause tells you how to "Share the good times" - it is an adverbial complement of "Share the good times".

No comma is required.