First of all, it is not a silly question at all, because a big part of being a religious Jew is believing in Moshiach, In fact it is one of the 13 principles of faith from the Rambam.
Now to get things straight we need to define two very common words that sometimes get mixed up.
Geulah – The era of redemption which we are waiting for.
Moshiach – The person who brings the Geulah to the world.
In the time of the Gemarah it was very popular for a Chasid to believe that his Rebbe is the Moshiach, so it would be easy to believe that the idea that ‘The Rebbe’ is Moshiach was something that was made up by some of the Chasidim, but that is not so.
It is the Rebbe himself who stated in his first Ma’amar that all the generations before us lead the way, and the job of our generation is to bring Moshiach himself. When the Brisker Rov (Rabbi Yitzchok Zeev Soloveitchik) heard that he said “דער יונגערמאן, מיינט אז ער איז משיח" (He thinks that he is Moshiach).
As the years passed, the Rebbe became more and more explicit saying many times that the Nasi of the generation is the Moshiach of the generation and he has no problem if people understand that he is Moshiach because this is the truth, that he is a prophet, and that his main prophecy is that we are the last generation of the Golus, and the first of the Geulah.
Later on, the Rebbe announced that Moshiach is already here, and we just need to open the door and usher him in the room, noting that big news happening then, such as the peace treaty between the U.S and Russia in regards to nuclear weapons, the falling of the Iron wall, and the Jews coming to Israel, are part of the Prophecies that our sages promised us and finishing off with saying that Moshiach’s name is Menachem! and 770 (the Rebbe’s synagogue is the place of the Shechina until the Geulah being a replacement for the Beis Hamikdash.
As a direct result the majority of the Chabad Chassidim believe that the Rebbe is the Moshiach that we are waiting for, and the main difference in opinions between the ‘Messianics’ and the ‘Anti-Messianics’ is to publicize or keep it as a known secret.
After ג' תמוז תשנ"ד, there was a lot of shock. And it was then that each Chabadnik had to make a very big decision. Should I believe what I just saw (The Levaya), or what the Rebbe and the Torah say (that he is Moshiach and Moshiach can’t die).
Now back to your question:
Did Chabad as an organization endorse this possibility or was this simply the opinion of devout individual followers of the man, Rabbi Schneerson?
Chabad as an organization is the Rebbe, and no one else can speak in the name of Chabad as a whole.
Did the Rebbe endorse this possibility? Yes.
P.S In case it is not obvious I am a Messianic, and I would be happy if someone from "the other side" would reply and explain what he thinks, or maybe mistakes I may have made.