Is it permissible for them to get married and stipulate that they will not have intercourse?

Firstly: 

It is not permissible according to sharee’ah for a man and
woman who are non-mahrams to live together in the same house. Hence likening
a couple living together without having intercourse to friends living
together is not proper. 

Both spouses must understand that one of the most important
aims of marriage is guarding and maintaining chastity and producing
offspring. This cannot be done without having intercourse. 

The Prophet (peace and blessings of Allaah be upon him)
encouraged marrying the woman who is fertile, and he told some of his
companions not to marry women who could not bear children. 

It was narrated that Ma’qil ibn Yasaar (may Allaah be pleased
with him) said: A man came to the Messenger of Allaah (peace and
blessings of Allaah be upon him) and said: I have found a woman who is of
good lineage and high status, but she cannot bear children; should I marry
her? And he told him not to (marry her). Then he came to him a second time
and he told him not to (marry her). Then he came to him a third time and he
told him not to (marry her) and said; “Marry the one who is loving and
fertile, for I will be proud of your great numbers.” 

Narrated by al-Nasaa’i (3227) and Abu Dawood (2050); classed
as saheeh by al-Albaani in Saheeh al-Targheeb (1921). 

See also the commentary on the hadeeth in the answer to
question no. 32668, and question no.
13492. 

But if the two spouses in the marital home agree not to have
intercourse, then this is something that could happen and it may be said
that it is permissible, in the case of spouses who are sick or elderly and
have no desire for intercourse. But if they have the desire for intercourse,
then how can they live together without being able to keep themselves
chaste? Where will each of them fulfil his or her desire if not with the one
with whom Allaah has permitted that?! 

It may also be possible and it may be said that it is
permissible if the woman is young and has desire, and she agrees to marry a
man who is impotent, has had his penis cut off or is elderly. The opposite
may also be said, which is that a man may marry a woman who is sick and has
no desire, or is physically incapable of intercourse, if he is able to be
patient and seek reward for that, or he has other wives with whom he may
fulfill his desires. 

Thirdly: 

The fuqaha’ distinguished between two issues with regard to
this matter: 

1 – When it is stipulated in the marriage contract that it is
not permissible for them to have intercourse. In this case the condition is
invalid, and the contract is invalid according to the majority of scholars. 

2 – When it is stipulated in the marriage contract that they
will not have intercourse. In this case it depends, but the most correct
view is that the contract is valid and the condition is invalid and carries
no weight and has no value, regardless of whether the condition is
stipulated by the husband, the wife or both. 

It says in al-Mawsoo’ah al-Fiqhiyyah (44/45): 

The fuqaha’ differentiated in the ruling on this stipulation
between two cases, when the condition denies that intercourse is
permissible, and when it stipulates that it will not be done. 

This is explained as follows:  

If it is stipulated in the marriage contract that intercourse
is not permissible, and that he married her on the basis that it is not
permissible for him, then there is no difference of opinion among the
scholars concerning the fact that this condition is invalid, but they did
differ as to its effect on the validity of the marriage contract. There are
two opinions: 

1 – The majority of Shaafa’i, Maaliki and Hanbali fuqaha’ are
of the view that both the condition and the marriage contract are invalid,
because this condition cancels out the purpose of the contract, and because
it contradicts it, because the marriage no longer makes any sense, rather it
is an unrealistic contract. 

2 –The Hanafi view is that the condition is invalid, but the
contract is valid, because the basic principle according to the Hanafis is
that marriage is not invalidated by an invalid condition, rather the
condition alone is invalid. 

But if it is stipulated in the marriage contract that there
will be no intercourse, the fuqaha’ differed concerning the ruling on that
and there are three views: 

1 – The Hanafis and Hanbalis are of the view that the
contract is valid but the condition is invalid. As for the invalidity of the
condition, it is contrary to the purpose of the contract, and it implies
waiving rights that would be binding according to the contract if this
condition were not stipulated. As for the contract remaining valid, that is
because this condition is something additional to the contract so it does
not invalidate it.  

The basic principle according to the Hanafis is that marriage
is not invalidated by an invalid condition, rather the condition alone is
invalid. 

2 –The Maalikis are of the view that the condition is
invalid, and the contract is invalid, because it has been done in a way that
is forbidden in sharee’ah. 

Then the Maalikis differed as to the consequences after the
contract is done. It was said that the marriage may be annulled before and
after consummation, or it was said that it may be annulled before
consummation and confirmed afterwards, and this condition is to be waived. 
This is the well known view of the (Maaliki) madhhab. 

3 – The Shaafa’is are of the view that if he marries her on
condition that he will not have intercourse with her, or he will only have
intercourse with her by day, or only once, for example, the marriage is
invalid if the condition was stipulated by her, because it is contrary to
the purpose of the contract. If it was stipulated by him, then it does not
matter, because intercourse is his right, and he may forsake it, and
enabling him to have intercourse with her is her duty, and she has no right
to forsake it. End quote.  

Fourthly: 

The woman should not agree to this marriage, and the man
should not agree with the woman if she does not want there to be any
intercourse between them. They should both understand that this is contrary
to sound fitrah (human nature). Allaah has created in men an inclination
towards women, and He has created in women an inclination towards men. There
are some people who direct their desires in haraam ways, and some who direct
them in halaal ways. Marriage is one of the laws of Allaah in which He has
permitted the man and woman to come together and has created affection and
compassion between them, and has brought forth children and offspring from
them. 

Allaah says (interpretation of the meaning): 

“And Allaah has made for you Azwaaj (mates or wives) of
your own kind, and has made for you, from your wives, sons and grandsons,
and has bestowed on you good provision. Do they then believe in false
deities and deny the Favour of Allaah (by not worshipping Allaah Alone)”

[al-Nahl 16:72]

“And among His Signs is this, that He created for you
wives from among yourselves, that you may find repose in them, and He has
put between you affection and mercy. Verily, in that are indeed signs for a
people who reflect”

[al-Room 30:21]

Marriage is the way of the Messengers (peace and blessings of
Allaah be upon them), and they are the best of mankind. Allaah says
(interpretation of the meaning): 

“And indeed We sent Messengers before you (O Muhammad
صلى الله عليه وسلم), and made for them
wives and offspring”

[al-Ra’d 13:38]

“At that time Zakariyya (Zachariya) invoked his Lord,
saying: ‘O my Lord! Grant me from You, a good offspring. You are indeed the
All-Hearer of invocation’”

[Aal ‘Imraan 3:38]

And Allaah knows best.