Project Cupid has officially opened on May 8th though there have been technical difficulties. (I’d expect as much!) At the moment it is only accepting applications for licenses. You have to fill out the form and schedule a virtual appointment with a clerk. The couple and witness must be physically located in NYC State.
You can then use a private officiant to marry you (virtual via video conference or with a socially distant wedding in person) and upload the signed marriage license directly to the witness and the marriage officiant. The officiant will then submit the signed license via email or the couple can upload it to their Cupid account. FAQ on using an officiant.
They are working on a video ceremony option with the clerks office but it isn’t ready yet. The licenses are normally valid for 60 days however on April 18, 2020, Governor Cuomo issued an Executive Order waiver that remains in effect “during the period of time that there exists a declared emergency in New York State.” I’d get the license and either hire an officiant (many are doing ceremonies via video conference now) or wait for the Project Cupid option.
The program is only valid until May 18th at the moment but I expect it will be extended. Follow the NYC Clerk on Twitter to ask questions or report issues. All participants signing in a video ceremony must be physically located in NYC State.