There isn't a Home Depot near me, the closest is about 1 hour away (one way). Lowe's is much closer and has most everything HD does. Lowe's has a veteran's discount card that I carry and they scan that. When I do go to Home Depot I just show them my driver's license which has "Veteran" on it and I get the military discount. I've not had anyone check for any kind of limit nor had anything excluded (that I'm aware of).
Many, many years ago, one of the stores was having a veteran's day sale with a discount for veterans. I took my DD214 in and they did not know what that was and declined to give me the discount. I prefer to have an id that is recognizable to the employees so the Lowe's card is nice, I shop there 1st and then HD if Lowe's doesn't have it.