Grace, I am not aware of any firm deadline to apply for LEED for any of the Design and Construction rating systems.
The issue is more likely to be whether you can successfully submit a LEED application after construction has begun, or after it has finished. Tracking down the necessary information at that point can be very challenging. It also is less useful—part of the point of LEED is to be a tool used for influencing design and construction.
For a project that is built, I would consider the LEED-EBOM rating system.