If a Muslim judge refuses to solemnize a woman's marriage without a legitimate Shar'i impediment, but rather due to laws that prevent it, and she has no Muslim guardian, is it permissible for a devout Muslim man to undertake her marriage?

It is permissible for a righteous man to marry a Muslim woman who has no guardian, or if it is difficult for the judge to marry her, even if a judge is present, due to the woman's need for it.

Conditions for the validity of the marriage:

Offer and acceptance by verbal declaration. Two Muslim, adult, sane witnesses who hear and understand that what was said in front of them is a marriage contract. Official documentation of the contract is not a condition for the validity of the marriage Islamically, but it is necessary for safeguarding rights.