No, you can't get married via Twitter.
Proxy Marriages used to be fairly common. There have been several famous proxy marriages throughout history, one of the biggest being the proxy marriage between Napoleon Bonaparte and Marie Louise, the Duchess of Parma. These days, however, proxy marriages are rare. There are only a few states in the union that will recognize them as being legal. In fact, most states, with the exceptions of Texas, Missouri, Colorado, California and Montana, most states consider proxy marriages to be illegal. Montana is the only state that allows a double proxy marriage. A lot of the time, the rules depend upon whether or not both parties need to be present to apply for the marriage license.
So what do you do if you want to get married but one of you cannot be there for the ceremony?
If you want to get married but there is exactly zero chance of one of you being present for the ceremony, the first thing that you need to do is talk to a lawyer. Your lawyer will help you figure out whether or not you meet your local exceptions to the proxy marriage laws. If he or she thinks you might have a case, your lawyer will talk to a judge who will grant permission (or not) for the ceremony to take place.
It is important to note that the reason you cannot be present for the ceremony is important. You will not be allowed to have a proxy marriage ceremony performed unless there are extenuating circumstances. Right now, just about the only reason that proxy marriages are allowed is if one of the parties is a member of the military who is on deployment.
It is also important that you understand that this is not one of those situations in which you can simply travel to the states in which proxy marriages are legal, have the ceremony performed and then go home. While a few states may recognize your marriage as legal under the "common law" rules, most will not. This is why it is better to work with an attorney and get an official sanction by a judge. It is that sanction that will help you ensure that your marriage is legally recognized everywhere.
If you are granted a proxy marriage, there are ways to make the ceremony more intimate. Video conferencing has opened up the doors to at least allow both parties to see each others' faces as they take vows. There is no reason that the marriage ceremony can't still be a good and heartfelt one.
Some people have tried to use proxy marriage to get around immigration laws. You should know though, that any proxy marriage that has not been consummated will not be legally recognized within the United States.
There are all sorts of reasons that could prevent you from being able to attend your marriage ceremony--in spite of your best intentions. This is why there are still some places and some ways that you can have a proxy marriage. Don't take this ability lightly, however, because it takes extenuating circumstances to get a legal proxy marriage. Make sure you contact an attorney before you try to get one on your own.