Is it permissible to take from the 10% designated "for the sake of Allah" which the husband committed to give from his salary, when there is an urgent need to buy basic household necessities? And is this commitment considered a vow?

Intrusive thoughts do not constitute a vow (nadhr). If the husband uttered it without the intention of making a vow, then he is not obligated to fulfill it; it becomes a promise that is commendable to fulfill. However, if he uttered it with the intention of making a vow, then it is considered a vow, and he is obligated to fulfill it when the condition upon which it is contingent is met, which is obtaining work abroad, based on the saying of the Prophet, peace and blessings be upon him: "Whoever vows to obey Allah, let him obey Him."

Regarding the amount of the vow, if the wording could mean donating a percentage of one month's salary or of every salary, then what matters is his intention at the time of making the vow. If he intended to donate it from one month's salary, then he may do so. If he intended to donate it from every salary, then there is a difference of opinion among scholars regarding this, and the most correct view is that he is obligated by what he intended if the wording allows for it.

As for the beneficiaries of the vow, if the husband did not specify a particular recipient, then the vow should be given to the recipients of charity, such as the poor and the needy. It is not permissible to give it to his household members for whom he is financially responsible.