Praise be to Allah.
If a woman’s guardian (wali) with regard to marriage is her cousin (son of her paternal uncle) and he wants to marry her, there is nothing wrong with that if she agrees.
Ibn Qudaamah (may Allaah have mercy on him) said:
The wali of a woman for whom it is permissible to marry her may be her cousin, her master, the judge or the ruler. If she gives him permission to marry her then he may do that. End quote.
In that case he may do the marriage contract on behalf of himself and the woman, because he is her wali. So he says: I marry you, or I marry myself to So and so, and so on. He does not need to say I accept, because his acceptance is implied. And she does not need to say I accept either, because a woman cannot do the marriage contract for herself or for anyone else, rather her wali does the marriage contract for her.
Or he may appoint another man to do his marriage contract on his behalf, whether this wakeel is a state official or someone else. In that case his deputy (wakeel) says: I give you So and so in marriage, and he says: I accept. Thus the marriage contract is done. Both ways are narrated from the Sahaabah (may Allaah be pleased with them).
Imam al-Bukhaari (may Allaah have mercy on him) said:
Chapter: If the wali is the suitor. Al-Mugheerah ibn Shu’bah proposed marriage to a woman when he was her next of kin (her guardian), so he told another man to do the marriage contract for him.
‘Abd al-Rahmaan ibn ‘Awf said to Umm Hakeem bint Qaarit: Will you put me in charge of your affairs? She said: Yes. He said: Then I offer to marry you.
‘Ata’ said: Let him bring witnesses, or let him order a man among her family. End quote.
Al-Albaani classed the reports of al-Mugheerah ibn Shu’bah and ‘Abd al-Rahmaan ibn ‘Awf (may Allaah be pleased with them) as saheeh in Irwa’ al-Ghaleel, 1845 and 1855.
It should be noted that it is essential to have the marriage contract witnessed in both cases. For information on the essential parts and conditions of marriage, please see the answer to question no. 2127.
And Allaah knows best.