I am a US permanent resident card holder who was born in Vietnam. My return flight requires an overnight (16 hour-non same day) transit in narita. I do not have a passport as I am traveling to vietnam under the passport except conditions of having a vietnam visa exempt as well as a green card. Both my flights from vietnam to narita and from narita to the US are with American Airlines. With that being said, am I allowed to transit in japan or will I be required to rebook.
The short answer is "no".
I am presuming that you are a Vietnamese citizen (even if you don't have a passport to prove it). Vietnamese citizens are allowed transit airside in Japan without a visa, but only if the departing flight is the same calendar day, which yours is not. Thus in order to fly on this itinerary you will require a transit visa for Japan, which will require a passport.
You MAY be able to obtain a reentry permit from the US, and Japan MAY issue a transit visa against that.
Even if you were to change your flights to have a same-day connection, it's not clear to me if you could proceed. Whilst the US and Vietnam may allow you to travel without a passport, Japan will likely not. Even in transit, your nationality matters as it is what controls whether you are allowed transit the country at all - and without a passport you do not have proof of your citizenship.
It's also worth noting that airlines will frequently have different requirements for accepting a passenger than the countries do, so even if the US, Vietnam and your transit country all allow you to travel without a passport or a reentry permit, the airline may still refuse boarding if you do not have one.
As an aside, whilst your flight from Narita to Vietnam may have an American Airlines flight number, it is almost certainly a codeshare with another airline - probably JAL - and not an actual AA plane.