Is what the husband did—using the company's stamp without the employer's knowledge to obtain a driver's license, despite the employer's refusal—unlawful (haram), and is it considered a betrayal of trust?

The general principle is that the employer does not have the right to prevent your husband from obtaining a driving license, as long as he possesses the ability to drive safely. This license is among the public freedoms guaranteed by Islamic law for every individual, in fulfillment of Allah's statement: {It is He who made the earth subservient to you, so walk among its regions and eat from His provision. And to Him is the resurrection} [Al-Mulk:15]. However, if the employer is obligated to bear and guarantee your husband regarding any damages that may result from his driving towards others, then he is permitted to prevent him, and it is forbidden to circumvent his permission.

If the employer is not obligated to bear the damages, then there is no blame upon your husband for what he did, and no responsibility on him. Otherwise, he would be committing a sin, and he must repent and stop driving until the employer permits him to obtain the license.