No. An inmate's account balance is treated as private financial information, the same way any person's bank account would be. Facilities do not make that information available to outside parties, including family members and significant others, without the inmate's explicit authorization.
The only person who can check the balance is the inmate themselves, and they can do so through whatever account access system the facility provides, typically at a kiosk, through their housing unit officer, or during commissary processing.
If you want to know whether a deposit went through correctly, you can call the deposit service you used, such as JPay or Access Corrections, and confirm on your end that the transaction was completed successfully. Whether those funds have cleared into the inmate's account and what the current balance is after other purchases is information only he has access to.