As mentioned by calccrypto it never becomes clear if Snape ever became a member of the original Order. He could have been spying for Dumbledore personally instead. Some known events can minimise the window in which he switched sides, but whether this made him a member of the Order is unknown.
Through Dumbledore’s pensieve, we are privy to his memory of the trial of Igor Karkaroff. Where Karkaroff accuses Snape of being a Death Eater, this was quickly countered:
Dumbledore had got to his feet. ‘I have given evidence already on this matter,’ he said calmly. ‘Severus Snape was indeed a Death Eater. However, he rejoined our side before Lord Voldemort’s downfall and turned spy for us, at great personal risk. He is now no more a Death Eater than I am.’
The Goblet of Fire, Chapter 30: The Pensieve
This seems to suggest that Snape at least took up his role as double agent prior to the day of Voldemort’s demise, 31 October 1981.
Snape would not have switched sides before Harry’s birthday 31 July 1980 as this is the earliest date for the subject of the prophecy to be determined.
After Harry’s birth, and after Voldemort deciding it was Harry and not Neville, would mark the earliest time Snape would have turned to Dumbledore for help.
The Harry Potter wiki gives us one possible other hint:
In October 1981, Dumbledore told the Potters that their best chance of remaining safe against Voldemort was the Fidelius Charm.
Moving the latest date to “October 1981”, as Dumbledore acted upon Snape’s advice here.
Thus Snape would have switched sides earliest 31 July 1980 and latest October 1981.
Edit to add an alternative timeline:
Cornelius Fudge retells the lead up to James and Lily’s hiding as follows:
One of [Dumbledore’s useful spies] tipped him off, and he alerted James and Lily at once. He advised them to go into hiding. […] Dumbledore told them that their best chance was the Fidelius Charm.
The Prisoner of Azkaban, Chapter 10: The Marauder’s Map
Followed a bit later in the conversation by:
Barely a week after the Fidelius Charm has been performed –
This suggests the timeline is much closer together. The Fidelius charm would have been performed on October 24 and even if it was ‘an immensely complex spell’ according to Flitwick I can’t imagine it taking more than a week to set-up. That would mean Snape went to Dumbledore around October 17.
This means Snape would not have switched sides until mid-October 1981. This left very little time for him to work as a double agent, which could explain why many members of the Order (e.g. Alastor Moody) stayed sceptical of him.
