Would I be committing a sin if I were to open a video game store where there is a possibility that youth and children might become addicted to gaming and neglect their religious duties?

If you adhere to the regulations for permissible games—namely, that they do not distract from an obligation and do not lead to something forbidden such as gambling, music, or pornography—then there is no harm in opening a game store, even if young people become addicted to them, as long as it does not lead them to something forbidden. It is preferable to add beneficial programs that guide patrons to what benefits them in their religious and worldly affairs.