What is the ruling on taking travel tickets granted by the employer for me, my wife, and my children, and is it obligatory to seek my wife's permission for her ticket?

If the intention is to take the ticket price in the event of the wife not traveling, then this does not depend on the wife's permission, but rather on the permission and condition of the employer providing the tickets. If the employer grants tickets for travel only, then you have no right to their price if travel does not occur. However, if they grant them as a right to the employee regardless of travel, then there is no harm in benefiting from their price. What is relied upon is the employer's condition and what was agreed upon, for the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions."