Not every couple holds a traditional wedding ceremony, particularly if the marriage is an arranged one, or the two members of the couple live in different places. It is possible to hold a marriage that meets local definitions of "legal" by proxy (in which someone with power of attorney stands in for the bride or groom), Skype or other electronic conferencing, or some other long-distance method. In order for a U.S. citizen to petition for (sponsor) a new spouse to get a U.S. green card and move to the United States, however, you'll need to make sure U.S. Citizenship and Immigration Services (USCIS) will recognize the union for green card purposes.
According to U.S. immigration law, a marriage where both persons are not physically present, often called a "proxy marriage," can be legally valid. However, the U.S. government will not recognize it as a basis for granting lawful permanent residence (a green card) unless the couple consummates the union after the marriage takes place; in other words, the couple has sexual relations. (This comes from § 101(a)(35) of the Immigration and Nationality Act or "I.N.A.")
"Pre-consummation" doesn't count in this context. And the U.S. adheres to the consummation requirement even if the marriage was valid under the laws of the place where it was held.
Demonstrating for certain that a marriage has been consummated is not something one could easily do without providing graphic or intimate material—and U.S. immigration authorities are not interested in or comfortable reviewing that.
Therefore, a simple affidavit or personal statement attesting to the fact that the relationship was consummated is an excellent start. Combine that with documentary evidence that both members of the couple were in the same place at the same time, such as copies of airplane tickets, hotel bills, photos taken together, and so forth.
Your affidavit will be more believable if you can explain that this is a continuation of a past relationship. As an added benefit, personal meetings before the marriage help show that the marriage is at real (bona fide). To avoid a finding that you are trying to commit immigration fraud, your credibility (believability) is key, so details and documentary evidence are all important.
If you've had a child together, that's a good indication that you have had sexual relations in the past. Nevertheless, U.S. immigration authorities will look only to the time period AFTER the wedding in making its determination as to whether the consummation requirement has been met. Children conceived before the marriage basically don't count for purposes of the consummation requirement.
If you've tried to have children but been unable to, medical records of visits to a doctor or fertility specialist, or a letter from the doctor or specialist, would be helpful to submit.
If travel would be difficult for one of you at this time, your better option might be to apply for a K-1 fiancé visa. This would allow the non-citizen to enter the U.S. for a 90-day stay, during which time you would get married, after which you could apply for a green card.
It's true that the immigrant would have to wait a little longer after entering the U.S. to start working—getting a work permit based on a fiancé visa can take weeks or months. After the marriage takes place, you'll need to submit the paperwork for adjustment of status (a green card), which will include another application for a work permit. But if you time things right and do some advance planning, this shouldn't take terribly long.
For more information and guidance on application procedures, see Fiancé and Marriage Visas: A Couple's Guide to U.S. Immigration, by Ilona Bray (Nolo).