Yes. When you successfully complete a contract and collect the auction earnings, the amount you won is added directly to your cash balance.
In government auctions, there is no cash system like in the Restaurant and Side Jobs sections. When you collect your earnings, the payment is transferred directly to your account and becomes available for use instantly.

What's the Difference from Restaurant and Side Jobs?
Contracts: When earnings are collected, the money is added directly to your cash balance.
Restaurant and Side Jobs: Earned income accumulates in the cash register first and must be collected from the register before it can be added to your account.
For this reason, since contract don't have an additional cash register system, you can access your payment instantly.
Important: After you successfully complete your contract and collect your reward, the amount you won is transferred directly to your cash balance without delay and can be used immediately in all your in-game transactions.