It is not permissible for a woman to observe i’tikaaf without her husband’s permission, because her observing i’tikaaf in the mosque affects the husband’s rights.
If he does give her permission, he has the right to cancel it and call her out of i’tikaaf.
Ibn Qudaamah (4/485) said:
The wife does not have the right to observe i’tikaaf without her husband’s permission… if her husband gives her permission, then he wants her to come out after she has started i’tikaaf, he has the right to call her out in the case of a voluntary i’tikaaf. This is the view of al-Shaafa’i… If what he gave permission for was something that she vowed to do, then he does not have the right to bring her out, because it becomes obligatory once it is started and must be completed, like Hajj once one has entered ihraam for it.
The Sunnah indicates that it is permissible for a man to prevent his wife from observing i’tikaaf without his permission.
Al-Bukhaari (2033) and Muslim (1173) narrated that ‘Aa’ishah (may Allaah be pleased with her) said: When the Messenger of Allaah (peace and blessings of Allaah be upon him) wanted to observe i’tikaaf, he would pray Fajr then enter his place of i’tikaaf. He ordered that his tent be pitched, intending to observe i’tikaaf for the last ten days of Ramadaan. Then Zaynab ordered that her tent be pitched, and others among the wives of the Prophet (peace and blessings of Allaah be upon him) also ordered that their tents be pitched. When the Messenger of Allaah (peace and blessings of Allaah be upon him) prayed Fajr, he saw the tents and said: “Do you intend righteousness by doing this?” Then he commanded that his tent be taken down, and he abandoned i’tikaaf in Ramadaan and observed i’tikaaf for ten days at the beginning of Shawwaal.
According to a report narrated by al-Bukhaari: ‘Aa’ishah asked him for permission (to observe i’tikaaf) and he gave her permission, then Hafsah asked ‘Aa’ishah to ask for permission for her and she did so.
“Righteousness” here means obedience or worship. Al-Qaadi said: He (peace and blessings of Allaah be upon him) said this by way of rebuke. He (peace and blessings of Allaah be upon him) had given some of them permission to do that, as narrated by al-Bukhaari. The reason why he criticized that was that he feared that they would not be sincere in their i’tikaaf, and all they wanted was to be close to him because of their jealousy concerning him, or because of his protective jealousy towards them, so he did not want them to stay in the mosque when it was the place where people gathered and was attended by the Bedouins and hypocrites, and they would need to go out and come in to attend to their needs and thus they would be exposed to people’s gaze… Or it may have been because he (peace and blessings of Allaah be upon him) saw them with him when he was in the mosque and it was as if he was at home with his wives, which was a distraction from the purpose of i’tikaaf, which is to keep away from one's wives and worldly concerns, etc. Or it may have been because they were crowding the mosque with their tents. This hadeeth indicates that women’s i’tikaaf is valid, because he (peace and blessings of Allaah be upon him) gave them permission, and he only stopped them from doing it afterwards for a reason. And it indicates that a man may prevent his wife from observing i’tikaaf without his permission. This is the view of all the scholars. If he has given her permission, can he prevent her after that? There is a difference of opinion among the scholars on this point. According to al-Shaafa’i, Ahmad and Dawood he has the right to stop her and bring her out of a voluntary i’tikaaf.
Ibn al-Mundhir and others said: This hadeeth indicates that a woman should not enter i’tikaaf until she has asked her husband’s permission, and that if she enters i’tikaaf without his permission he has the right to bring her out; if he has given permission he has the right to change his mind and stop her. It was narrated from ahl al-ra’y that if the husband gives her permission then stops her, he commits sin thereby and she may refuse. It was narrated from Maalik that he does not have the right to do that. But this hadeeth is proof against them.
From Fath al-Baari..