I get 1 or 2 of these a month and I have one in front of me now. It says:
This offer is limited to a one-time award for new members of a $200 Disney gift Card. and First Time cardmembers only. Some people have applied multiple times and claim that Chase/Disney doesn't care. The only problem with that is that they can't provide a screenshot or a link to any Chase or Disney policy that says you can apply as many times as you like whereas the policy I just quoted is spelled out right on the application.
So unless someone can point me to something official that says otherwise I interpret that to one application for each family member you can manage to get approved: yourself, your spouse, children over 18, etc. A joint membership wouldn't qualify because one or both of the joint qualifiers would have already received a gift card as a single applicant.
I suppose that if you needed $200 that badly you could apply, get a gift card, cancel the card after a while and reapply later. How this would affect your credit rating is anybody's guess.
You have to get a letter with a code.