Generally, states have background check processes for school employees. Certain felonies, particularly those involving violence, child abuse, or sexual offenses, will almost certainly disqualify an applicant. Other felonies might be considered on a case-by-case basis, depending on factors like the severity of the crime, the time elapsed since the conviction, and evidence of rehabilitation. Some states might have specific laws addressing this, while others leave it to the discretion of the hiring authority.
In short, a felon *could* potentially work for a public school, but it's far from guaranteed and requires a careful examination of the individual's record and the relevant state and local laws.