It depends on what you’re seeking. If you’re seeking a criminal case against Walmart, it depends on your state’s specific gun laws. Some states have really lax gun laws, while other states have more stringent requirements that mandate certain types of background checks. As you may have heard in the news, the federal government is having trouble passing a law that would require mental health and strict background checks in order to purchase guns. Whether it is a crime to sell a gun to someone who is mentally ill is a question you have to ask a lawyer in your jurisdiction, as the laws in many states vary considerably.
In California, if you have been admitted to a mental health facility and have been adjudicated to be a danger to yourself or to others, you cannot be allowed to own a firearm or a dangerous weapon. Any person who knowingly supplies, sells, gives or allows possession of a firearm to a person who has been adjudicated to be a danger to self or others shall be punished by 2, 3, or 4 years in prison (see California Welfare and Institutions Code section 8101). Therefore, if Walmart knew that your daughter was mentally ill and they knew that she had been adjudicated as a danger to herself or others, then they would criminally liable for selling her a firearm. Absent this specific knowledge, however, Walmart would not be liable under this section of the statute.
If you want to pursue a civil action against Walmart, say for example, a wrongful death suit, then you would need to speak with a personal injury attorney who can properly advise you.