When a woman does not have a wali

The asl in regards to a woman who does not have a wali is that the Islamic judge carries out her marriage by acting as her wali.

Ibn Qudaamah (paraphrase of what he) said: We know of no difference among the scholars regarding the fact that the Sultaan is the wali for the woman who has no wali and contracts her marriage. And the Imaam here refers to the Imaam, haakim, etc.

al-Mughni Volume 9 Page 360

When a woman lives in a place where there is no Islamic judge, she is allowed to entrust her affair to a just Muslim man so that he may contract her marriage.

Ibn Qudaamah (paraphrase of what he) said: If there is no wali for the woman and nor is there any Sultaan, then it has been narrated from Ahmad what suggests that a just man can marry her off by her permission.

al-Mughni Volume 9 Page 362

al-Qurtubi (paraphrase of what he) says: When a woman lives in a place where there is no Sultaan and she has no wali, then her matter (regarding her marriage) goes to a person who is reliable from among her neighbors. He marries her off and is her wali in this circumstance because people are always in need of getting married and they act in regards to this matter of marriage by the best means possible.

al-jaam'i lii 'ahkaam al-Qur'aan
Volume 3 Page 76

al-Nawawi (paraphrase of what he) says: Yunus bin abd al-'a'laay narrated that al-Shaafi'i said: It is permissible for a woman who lives in a place where she has no wali to entrust her affair to a man who marries her off.

Rawdah al-taalibiin
Volume 7 Page 50

The above holds when there is no Sultaan. If a woman has no wali but there is a Sultaan, then the scholars have differed as regards to if she can entrust a pious just Muslim man to marry her off.