Is it permissible to marry an Asian polytheistic girl after she has declared her Islam by saying, "I have given myself to you in marriage," without documentation, due to the impossibility of that because of illegal residency and the impossibility of affording the costs of marriage?

Marriage is not valid by merely saying "I marry myself to you" and the other party accepting. Rather, there must be a legal guardian for the woman, an offer from him, an acceptance from the husband, and the presence of two just witnesses. In the absence of a legal guardian, the Islamic judge (Qadi) marries her; and if there is no judge, then a just Muslim man. Marriage in this form does not require official declaration, but if the marriage is intended to be dissolved by divorce once circumstances improve, then it is a disputed marriage. However, if the intention is to declare or officially document it later, then there is no harm.