Does my ex-husband, who is military, have any grounds to request that our son stay with his stepmom over summers that he could possibly be deployed?
He has grounds to ask but you also have the opportunity to request full custody if you do not currently have it. The issue will be regarding the existing order and what it currently indicates and whether his commanding officer has certain pre-existing plans in place for child care and custody if he does get deployed. If this is a temporary visitation issue, and if you do not approve or wish for this to occur, you will need to respond to any motions to amend the order.
Given the factual grounds for the request your "ex" can seek such but legally he cannot unless the son wants to be with his stepmother as opposed to you over the period of time that you have written about. Given your situation I suggest that you consult further with a family law attorney about such.