USCIS (United States Citizenship and Immigration Services) does recognize the legal validity of a proxy marriage immediately upon its solemnization (officiation). Notwithstanding, UCIS will require that the marriage be "consummated" before it may be recognized for immigration purposes. This is known as the "Consummation Requirement."
In order to satisfy the "Consummation Requirement," a couple, who has been married by proxy, must prove to UCIS that the couple has been physically together after the proxy marriage was solemnized (officiated); moreover, the couple may do so by providing copies of visas, airline tickets, photographs, etc.
A bride or groom, who is a citizen of a foreign nation and who has entered into marriage by proxy, but who has not yet consummated the marriage, may apply for a K-1 Visa by submitting an I-129F, as well as a G-325A for each her or himself as well as for the sponsoring spouse. Conversely, upon consummating the marriage, such a bride or groom may apply for a K-3 Visa by filing an I-130 instead of the I-129F.
Notwithstanding the caveat in regards to USCIS' "Consummation Requirement," a proxy marriage is recognized as a legal marriage by all branches of the U.S. Federal Government, all U.S. States and territories, and all foreign nations which recognize U.S. marriages.
The U.S. Armed Forces have no "Consummation Requirement." The U.S. Armed Forces allow a spouse, who is married by proxy, as well as any children of such a union (and even of a prior union, with certain conditions), to be enrolled in DEERS immediately upon the solemnization of the marriage, even before the marriage is consummated. This allows the non-enlisted spouse and any children to immediately start to receive military marriage benefits, including TRICARE, married or family BAH (Basic Allowance for Housing), Family Separation Allowance, entry to the Commissary and the Exchange, Educational Benefits, Death Benefits, etc., etc..