Simple question: discounting Dumbledore (who knew Snape's loyalty was with him) and, eventually, Harry, did the other members of the Order of the Phoenix like Sirius, Remus, Hagrid and others know that Snape was on their side?
Before the Battle of Hogwarts: No. I have no hard source for this, but there is some evidence:
- Snape was treated like a traitor, for example, they secured Grimauld Place 12 against him
- The Order forced Snape out of Hogwarts shortly before the final battle, that's not something you would do to an ally in such a desperate situation
- After the battle at the Astronomy Tower, everyone seemed surprised by Snapes actions, so they were not aware of Dumbledore's plan to get killed by Snape, and thus had no reason to still expect him to be on the Orders side.
Snape was always known to be a double-agent, but everyone trusted his loyalty being with Dumbledore. At least, until the Battle of the Astronomy Tower.
[...] Sirius and Snape were eyeing each other with the utmost loathing.
"I will settle, in the short term,' said Dumbledore, with a bite of impatience in his voice, "for a lack of open hostility. You will shake hands. You are on the same side now. Time is short, and unless the few of us who know the truth do not stand united, there is no hope for any us."
Very slowly — but still glaring at each other as though each wished the other nothing but ill — Sirius and Snape moved toward each other and shook hands. They let go extremely quickly.
"That will do to be going on with," said Dumbledore, stepping between them once more. [...]
Harry Potter and the Goblet of Fire, chapter 36, "The Parting of the Ways"
Here Dumbledore made absolutely clear to Sirius, a prominent member of the Order of the Phoenix, that he trusts Snape.
"It isn't our business to know," said Lupin unexpectedly. He had turned his back on the fire now and faced Harry across Mr. Weasley. "It's Dumbledore's business. Dumbledore trusts Severus, and that ought to be good enough for all of us."
"But," said Harry, "just say — just say Dumbledore's wrong about Snape —"
"People have said it, many times. It comes down to whether or not you trust Dumbledore's judgement. I do; therefore, I trust Severus."
Harry Potter and the Half-Blood Prince, chapter 16, "A Very Frosty Christmas"
Here Lupin makes clear that he trusts Snape because he trusts Dumbledore. A bit further in the same chapter, he makes clear that Snape helped him when he was teaching Defense Against the Dark Arts at Hogwarts.
The Battle of the Astronomy Tower changes this.
"Snape," repeated McGonagall faintly, falling into the chair. "We all wondered... but he trusted... always... Snape... I can't believe it. ..."
"Snape was a highly accomplished Occlumens," said Lupin, his voice uncharacteristically harsh. "We always knew that."
"But Dumbledore swore he was on our side!" whispered Tonks. "I always thought Dumbledore must know something about Snape that we didn't. ..."
Harry Potter and the Half-Blood Prince, chapter 29, "The Phoenix Lament"
Of course, after Snape killing Dumbledore, how could anyone still believe him not to be on Voldemort's side?
Only through what Harry learns of the deal between Snape and Dumbledore, is Snape redeemed again at the end of the saga.