Is the wife's utterance of the two testaments (Shahadatain) in front of her husband considered valid and sufficient for her conversion to Islam, knowing that she embraced Islam on her own and learned about it via the Internet, and the marriage took place in the mosque in the presence of witnesses, and she obtained a certificate of her conversion to Islam?

A person's declaration of Islam is not a condition for their conversion; it is a matter between the servant and their Lord. However, if it is declared for documentation, there is no harm in it. As for marriage, it is not valid without a guardian for the wife, and a disbeliever has no guardianship over a Muslim woman. In the absence of a Muslim guardian, the judge, the imam of the mosque, or the regional mufti acts as the guardian.