In this day and age, some people think it's okay to just use a site like evite for invitations, etc... so you could go this route, since email addresses are what you were provided.
However, if you feel it is not the proper way to go, you can extend the invite via phone call to each person (I like what talkstotrees suggested), perhaps as a "save the date", and then get their address and mail the formal invite as soon as you have the registry info (or don't wait for the registry info, since some consider it tacky to put that into the invite).
BTW, I hosted a baby shower where I was given email addresses for everyone, and since it was super short notice (someone else was supposed to host and dropped the ball), I used evite. The Mom-To-Be's Aunt refused to come to the party because the only invite she got was via email (not that she didn't get it in time, just that it was the only form of invitation). For the following baby shower (a few years later) I made sure to do mailed invites (though the Aunt still didn't come).
ETA: I wouldn't recomment sending a text with instructions to replay via email, because I've learned that people just don't always pay that close of attention... and will reply via text (putting you in a whole lot of hurt). Just my 2cents.