According to an article in Bride's "Zoom Weddings Are Now Legal!" dated April 2021,several states including California, New York, Ohio, and Colorado have made temporary arrangements to make it legal for weddings to take place over video conference in response to the pandemic. Several also permit remote applications for marriage licenses.
According to the June 2021 article "Is it legal to have a virtual wedding?" from Wedfuly states permitting such weddings include New York, California, and Illinois. However, in Florida, Hawaii, Ohio, Tennessee, and Texas the article says that both parties to the wedding, and the officiant must all be physically present in the same place, although the wedding may be displayed on video for people at other locations.
According to the New York Times article "Why Virtual Weddings Are No Longer Legal in New York" dated July 16, 2021, on June 25, 2021 Gov. Andrew M. Cuomo ended the executive order he originally issued in April 2020 allowing couples to be married online.
According to that NYT article new legislation would be required for such weddings to be legal in New York again.
Other states might cancel or change their temporary rules allowing "virtual" weddings at any time. The New York decision apparently came as a surprise, causing a number of scheduled weddings to be canceled or changed to in-person events. One planning such a wedding would do well to check current rules in the state involved.
Citizenship or permanent residence should not matter to the validity of such weddings, as it does not matter for in-person weddings. However, it seems that both parties and the officiant must all be physically present within the same US state or other jurisdiction for the marriage to be valid.
Not all countries recognize the validity of such "virtual" marriages. It is specifically mentioned that the UK does not.
Also, it seems that while such weddings were lawful in New York, the temporary rules required that both members of the couple, and the officiant, be physically within the state of New York for the wedding to be legal. Other states seem to have similar rules.
If the members of the couple are not in the same state, it seems that the wedding becomes a "proxy wedding". This is a process where one person is not present at all, but is represented by a "proxy". Proxy weddings have been lawful far back in history, at least as far as the middle ages in Europe, and perhaps farther. They are currently lawful in only a few US states. According to the Wikipedia article linked above these are: Texas, Colorado, Kansas, and Montana. Such weddings are invalid in all other states. Some of those states limit proxy marriages to active members of the US Military on deployment.
The service Web Wed seems to claim that members of a couple may be located anywhere or the virtual wedding, but this appears incorrect. They also claim a patent on some forms of this process, and seem to say that they are the only legitimate provider of virtual weddings, but the articles linked above quote various officiants and services that have performed hundreds of such marriages.
Under the "full faith and credit" clause of the US Constitution, marriages that are legal under the laws of any US state must be recognized by all other US states. Other countries may or may not recognize such marriages, depending on their own laws.
I do not find any court case in which such a wedding was challenged and a court ruled on its validity, one way or the other.
An answer by user Roberto says that Web Wed does this by having an officiant in Utah County, Utah. The Utah County Clerk's office page on Web Ceremonies does say that the counter permits weddings by video conference as long as the "host location" is in the state of Utah. However, it does not mention WebWed specifically, and some of the information does not match. WebWed's site says:
Once you have the license, there is no rush to perform the ceremony. Once issued thru the courts the marriage licenses never expire, so once you have it you can get legally married whenever you wish. However, please note that each state expiration is different. You must adhere to that specific state policy. In most states, you have 30 days to get married after the license is issued.
Utah marriage licenses are valid for only 30 days and are not issued by a court, so that does not seem to be what WebWed is talking about. The public pages of their site do not seem to mention Utah specifically.
The WebWed site also says:
A state issued marriage license is required to use this service. Please note that it is the registers/users responsibility to verify and abide by state, county, and armed forces marriage laws prior to using this service.
WebWed's site advises government officials to reject any online marriage not preformed through their service. If such marriages are lawful and valid, they will be available to anyone that follows the proper rules. US states do not grant exclusive rights to specific commercial services in such cases.
US Federal Law
8 U.S. Code § 1101 which is the Definitions section for CHAPTER 12— IMMIGRATION AND NATIONALITY, says that:
(a) As used in this chapter—
(a) (35) The term “spouse”, “wife”, or “husband” do not include a spouse, wife, or husband by reason of any marriage ceremony where the contracting parties thereto are not physically present in the presence of each other, unless the marriage shall have been consummated.
"consummated" is not defined in this section, and so presumably has its normal meaning in regard to marriage, that is, that the parties have had sexual relations at least once after the marriage occurred.
This section would apply to proxy marriages as well as to "virtual" marriages, but applies only in the context of immigration law.
Normally US Federal law leaves the issue of marriages to the states, except where there is a claim that Federal constitutional rights are being infringed, such as in the cases of Loving v. Virginia, 388 U.S. 1 (1967) which forbade states to prohibit interracial marriage, and Obergefell v. Hodges, 576 U.S. 644 (2015) which forbade states to prohibit same-sex marriage.
Section 30-1-6 of the Utah Code provides that:
30-1-7. Marriage licenses -- Use within state -- Expiration.
(1) No marriage may be solemnized in this state without a license issued by the county clerk of any county of this state.
(2) A license issued within this state by a county clerk may only be used within this state.
(3) A license that is not used within 32 days after the day on which the licensed is issued is void.
Note that 30-1-6(2) says that any such marriage shall take place "within the state". Nothing in this section, or any other section of chapter 30, specifically authorizes a marriage by video conference where one or both parties are not within the state when the marriage occurs. It is not clear on what basis the Utah County clerk's office considers such marriages valid.
Section 30-1-8 of the Utah Code provides that:
(b) Each applicant and if an applicant is a minor, the minor's consenting parent or legal guardian, shall appear in person before the clerk and provide legal documentation to establish the following information ...
However that may be relevant only when one or both parties to the marriage is a minor.
I can find no other provisions of the Utah code which control applications for a license by web, or specifies where a marriage may or may not be held, or who is or is not required to be physically present at a valid marriage ceremony.