Final answer:
The National Voter Registration Act of 1993, also known as the Motor Voter Act, is the legislation that mandates states to offer voter registration at welfare and driver-licensing offices.
Step-by-step explanation:
The legislation that required states to make voter registration available at welfare and driver-licensing offices is the National Voter Registration Act of 1993, commonly known as the Motor Voter Act.
This act was designed to facilitate the voter registration process and ensure that more citizens would have the opportunity to register to vote, thereby promoting greater participation in the democratic process.
It requires states to provide individuals the option to register to vote when they apply for or renew their driver's license and at offices that provide public assistance.