Is it permissible to marry عرفياً ( عرفي - a type of informal marriage contract in some Muslim countries without official registration, but in compliance with Islamic law, where the contract is written and signed by the couple and two witnesses) with two witnesses and a dowry for the purpose of travel and preserving the wife's religion, knowing that her guardian is not a Muslim?

Online interaction between men and women is impermissible because it leads to fitna (temptation/strife) and corruption. Residing in non-Muslim lands entails significant risks, and marrying this woman is not a necessity. If she is in a place where Muslims are few and there is concern for her deen (religion), then she must emigrate (hijra). A woman's travel without a mahram (unmarriageable kin) is not permissible except in cases of dire necessity. Marriage is not valid unless its conditions and pillars, such as the sigha (pronouncement of offer and acceptance) and the wali (guardian), are met. Marriages resulting from online interactions often end in failure, so you should sever the relationship and seek a righteous woman from your own country.