Ricky Roma had not officially won when he took down the Cadillac poster. He did this as an act of bravado. It is the manager Williamson (Kevin Spacey), as an representative of Mitch & Murray, who would be the person with the authority to announce a winner.
You have to keep in mind not only the personality of Roma but the incredible stress the salesmen ...
If Roma didn't make the sale until after Williamson left the office,
how could Williamson have filed the contract?
At 24m when closing the office Williamson says, "I got to go home, grab one hour with my kids, come back here, see if any of you closed a contract, take the contract to the bank..." Much later, at around 1h27m, he also says, "...
This is a massive spoiler:
Levine and Williamson are in the car. Williamson leaves.
Roma closes the deal, goes to the office and lets the contract at the office.
The joint gets ripped off, and the burglar steals some of the contracts.
In the morning, Williamson lies to Roma saying that some contracts have been stolen but not Lingk's ...