**The Rebellion probably did not know that the Emperor was Darth Sidious. That appears to be the intention of George Lucas and the [Lucasfilm Story Group](http://starwars.wikia.com/wiki/Lucasfilm_Story_Group), at least.**

The Emperor took great pains to make sure the public did not know his Sith identity. [He did not use his Sith name publicly for political reasons](https://scifi.stackexchange.com/a/92291/31936), and almost no one saw the Emperor use his power:

>  [The Emperor] removed his robe, took the elaborately crafted hilt of his lightsaber in hand, and ignited the blade. Vader looked on in surprise. He seldom saw his Master so publicly demonstrate his power. And he understood what it meant, of course. There must be no survivors who could bear witness. Only the Royal Guards could be allowed to live—only they could be trusted never to reveal what they’d seen, or even to talk about it among themselves.
>
> <sub>**[*Lords of the Sith*](http://starwars.wikia.com/wiki/Lords_of_the_Sith), p. 168**</sub>

The fact that the Emperor took such pains to hide his Sith identity even from the vast majority of *Imperial* soldiers suggests that the Rebellion didn't know, either. Even [Grand Moff Tarkin did not know for sure that Palpatine was a Sith Lord](https://scifi.stackexchange.com/a/122341/31936)  -- he only suspected, despite being on a first name basis with Palpatine.

That said, you bring up an excellent point that Bail Organa might easily have found out from Yoda or Obi-Wan, and if so Organa would probably have told the rest of the Rebels since he was one of their leaders. The [draft script of Episode III on IMSDB](http://www.imsdb.com/scripts/Star-Wars-Revenge-of-the-Sith.html) does include a mention of Darth Sidious' name in Organa's presence:

> Bail Organa: My wife and I will take the girl. We've always talked of adopting a baby girl. She will be loved with us.
>
> Obi-wan: And what of the boy?
>
> Yoda: To Tatooine. To his family, send him.
>
> Obi-wan: I will take the child and watch over him. Master Yoda, do you think Anakin's twins will be able to defeat Darth Sidious?

However, Sidious' name is *not* mentioned in the actual movie (the [same scene can be seen on Youtube](https://www.youtube.com/watch?feature=player_detailpage&v=a_VKYf89XEE#t=149)), perhaps because the filmmakers realized Organa couldn't have known about Sidious or else the Rebel Alliance would have.

No one in the Rebellion seems to have known that Palpatine was a Sith Lord in Episode IV because, out-of-universe, [even George Lucas didn't know the Emperor was Darth Sidious](https://scifi.stackexchange.com/a/102390/31936). The Emperor's Sith affiliation wasn't revealed until Episode V. By that time Luke had met Yoda, and Yoda could have told Luke who in turn would have told the rest of the Rebellion.

This leaves us to explain why Bail Organa apparently *didn't* know the Emperor was Darth Sidious despite the fact that he rescued Yoda from Sidious. The best explanation I can come up with is that Yoda and Obi-Wan decided not to tell Organa because of one or more of the following reasons:

 * the Emperor's Sith identity was Jedi/Sith business, so Organa didn't need to know
 * Organa already opposed the Emperor, so Yoda and Obi-Wan wouldn't really gain anything from telling him
 * it was better for Organa not to know since Organa was raising Vader's daughter Leia yet still worked in close proximity to the Emperor (because he was an Imperial Senator). Telling Organa could increase the risk that the Emperor would find out about *Leia's* true identity.