What is the ruling on young people agreeing, when creating an email account (especially on Yahoo), to the condition of arbitration under the law of a state that does not rule by what Allah has revealed, in the event of problems with the institution owning the email service?

If the condition entails recourse to a law that contravenes Sharia, then it is not permissible to agree to it. However, if it does not entail that, then there is no harm in agreeing to it; for it is among the permissible conditions that Sharia has allowed, based on the saying of the Prophet, peace and blessings be upon him: "Muslims are bound by their conditions, except for a condition that prohibits what is lawful or permits what is unlawful."