Is it permissible to propose to a woman with whom the marriage contract was done but her husband is missing and they do not know anything about him?

Firstly:

In the answer to question no. 129851 , we stated that the marriage contract should be registered officially, so as to protect people’s rights and protect this important contract from being toyed with. This case is an example of the problems that may result from not registering the marriage.

Secondly:

If the marriage contract has been done with a woman, then she becomes married, even if the marriage contract was not registered officially. It is not permissible for anyone to propose marriage to a married woman, no matter what the reason, whether her husband is absent or present. A stern warning concerning that was narrated from the Prophet (blessings and peace of Allah be upon him), who said: “He is not one of us who turns a woman against her husband.” Narrated by Abu Dawood (2175); classed as saheeh by al-Albaani in Saheeh Sunan Abi Dawood.

It says in al-Fataawa al-Kubra by Ibn Taymiyah (2/266):

In the case of a married woman, it is not permissible to propose to her, either explicitly or implicitly; rather that comes under the heading of turning her against her husband, which is one of the most abhorrent of sins. End quote.

Thirdly:

With regard to the absence of her husband, if it is not possible to find out where he is and get in touch with him, then he is described by the scholars as missing. The ruling regarding his wife is that the judge should set a time limit for her, then if this time limit expires without finding any trace of her husband, in that case the judge should rule that he has died, and his wife should begin to observe the ‘iddah of one whose husband has died, starting from the time when the judge rules that he has died. Then after her ‘iddah ends, she may remarry if she wishes.

The one to define that time limit is the judge; the wife has no right to define that time limit herself.

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in ash-Sharh al-Mumti‘ (13/374):

The view of the three imams (Abu Haneefah, Maalik and ash-Shaafa‘i) is that it is essential to refer to the judge, and he is the one who should take charge of this matter. This is what must be done, especially according to the more correct scholarly view, which is that reference should be made to the judge to decide whether to assume that the husband is dead. But we may say that the ‘iddah following the death of the husband does not need the verdict of a judge; once the judge has set a time limit (after which it is to be assumed that the husband has died), that implies that when that time limit expires, the woman should begin to observe the ‘iddah of one whose husband has died, and there is no need for the judge to pass a verdict concerning  that. End quote.

So what you must do is leave this woman alone and forget about her, starting immediately. The matter of her husband’s absence is to be left to her and her family to decide whatever they want; either she made choose to wait for him or she may choose to refer the matter to the judge.

And Allah knows best.