You said "sent out a message"--did you directly send a messages individually to each of these people?
Or did you do a group posting of some kind?
I vote for a careful, disciplined gathering of all their contact info, yes, even if it means you have to call them. Some people put their phone numbers on their Facebook profile, or their personal emails.
Start there, and contact them one at a time. And build your address book that way.
And feel free to ask if they'll give you other people's addresses. Theoretically they wouldn't without permission, but lots of people ignore that little piece of etiquette, and you might as well make that work for you.
I think this goes both ways--you want people to seem eager to get the invitation, and they might want you to seem eager to gather the info. If you do something sort of "junk mail" seeming, they may not take you too seriously.
But then again, maybe you did contact them individually.
And if you did, and they don't bother to click "reply" and type in their address, then yes, I'd think they don't want to come. Feel free to message them again and say, "Remember when I asked you for your address so I could send you a wedding invitation? I haven't seen any reply, so I thought I'd try one more time. If I don't hear back, I'll assume you don't want to come." Though that seems whiny, doesn't it? Maybe "If I don't get your address, I won't be able to address your invitation."