Can an employer fire you due to an unacceptable background check if you have already been hired?

Most employment relationships are what is known as "at will".  This means that you can work for an employer, or not, your choice.  In turn, the employer can hire or fire someone for any reason or no reason whatsoever.

The only exceptions to this are: 1) If there is an employment agreement or contract (which governs termination); 2) If there is a union or collective bargaining agreement (which governs termination); 3) If there is an employee handbook or other official policy statement (which governs termination); 4) If discrimination is a factor (there can be no termination for reasons of race, religion, age, disability, sex, national origin).

As for letting someone work with vulnerable adults prior to completion of the background check, this is certainly questionable and potentially dangerous practice.  Had an incident occurred as a result they could have opened themselves up to tremendous liability.  But as it stands, you have no claim for such negligent conduct.