She has a defect that prevented her from getting married, but she wants to marry a man so that she may be artificially inseminated with his sperm, then she will ask him to divorce her

Firstly: 

If what you mean by adoption is taking a child from
someplace, such as an orphanage for example, in the sense that you will
sponsor a good upbringing for him in accordance with Islamic teaching, take
care of him, teach him that which will benefit him in both religious and
worldly terms, and spend on him, then this is permissible. In fact it is one
of the righteous deeds and acts charity that are encouraged by Islam,
especially if this child is an orphan. We have discussed the virtue of
sponsoring orphans in fatwa no. 188161. 

However we should point out that if the child is male and has
reached the age of puberty – or even if he is approaching puberty – then you
must interact with him as you interact with any other non-mahram. So you
should not be alone with him, or appear in front of him with your adornments
(i.e., without hijab), and he does not inherit from you after you die, but
it is permissible for you to bequeath part of your wealth to him, to a
maximum of one third of the estate. 

But if what you mean by adoption is taking a child in the
sense that the same rulings become applicable to him as to one’s own child,
and after he reaches puberty you allow him to interact with you like a
mahram – if the child is male – and you appear in front of him (without
hijab), and he can be alone with you and travel with you, and he inherits
from you after you die, then this is definitely haram. Allah has forbidden
adoption in the Holy Qur’an, as He says (interpretation of the meaning):

“Allah has not put for any man two hearts inside his body.
Neither has He made your wives whom you declare to be like your mothers
backs, your real mothers. (AzZihar is the saying of a husband to his wife,
“You are to me like the back of my mother” i.e. You are unlawful for me to
approach.), nor has He made your adopted sons your real sons. That is
but your saying with your mouths. But Allah says the truth, and He guides to
the (Right) Way.

Call them (adopted sons) by (the names of) their fathers,
that is more just with Allah. But if you know not their fathers (names, call
them) your brothers in faith and Mawaleekum (your freed slaves). And there
is no sin on you if you make a mistake therein, except in regard to what
your hearts deliberately intend. And Allah is Ever OftForgiving, Most
Merciful”

[al-Ahzaab 33:4-5]. 

Hence it is not possible – according to the law of Allah, may
He be glorified – for this child to come under the same rulings as one’s
real child. 

Secondly: 

Getting married with the intention of divorce is haraam and
is not permissible. Marriage is a solemn covenant which Allah, may He be
glorified, calls in His Book a firm and strong covenant, as He says
(interpretation of the meaning): “And how could you take it (back) while
you have gone in unto each other, and they have taken from you a firm and
strong covenant?” [an-Nisa’ 4:21]. It is not permissible to take a
covenant of this nature as a focus for tampering or whims and desires. We
have previously discussed marriage with the intention of divorce in fatwa
no. 111841 

Based on that, it becomes clear to you that it is not
permissible for you to marry this man with the intention of divorcing him
once you have fulfilled your aim of getting pregnant, because this temporary
nature of the marriage will either be mentioned in the marriage contract, in
which case the marriage contract will be invalid according to scholarly
consensus, because it will have become a contract for mut‘ah (temporary
marriage), which is haram according to scholarly consensus, as we have
explained in fatwa no. 6595; or, you will not clearly
state the temporary nature of the marriage in the marriage contract, instead
concealing it in your heart. This is also not permissible, because it is
deceit. Moreover, if we assume that this marriage takes place, how will you
free yourself from this marriage after you have got what you want? You will
have no choice but to ask for talaaq or khula‘, both of which are not
allowed without a genuine, shar‘i reason, as we have explained in fatwa no.
34579 and 176224. 

Thirdly: 

Stipulating that there should be no intercourse in marriage
is not permissible, because that is a condition that is contrary to the
purpose of marriage. Hence it is not permissible, it is not valid and it
does not have to be fulfilled. In fact some of the scholars regard this
condition as invalidating the marriage contract altogether. It says in
al-Mughni by Ibn Qudaamah (7/95): If it is stipulated that he should not
have intercourse, that could render the contract invalid, because it is a
condition that is contrary to the purpose of marriage. This is the view of
ash-Shafaa‘i. End quote. 

In Mughni al-Muhtaaj (4/377) it says: If a condition
is contrary to the basic purpose of marriage, such as a condition that the
husband will never have intercourse with her at all, … or that he will
divorce her even after having intercourse, the marriage contract becomes
invalid, because that is contrary to the purpose of the contract, hence it
renders it invalid. End quote. 

We have previously discussed this issue and the differences
of scholarly opinion concerning it, in fatwa no. 94454. 

The way in which you want to marry this man is not
permissible, so you should forget about it altogether, seeking the pleasure
of Allah; be patient in bearing this trial with which you are being tested
and seek reward for your patience with Allah, may He be glorified. You
should also look for a way to treat this birthmark and offer a lot of du‘aa’,
asking Allah to grant you healing and well-being, and to make things easy
for you. We ask Allah, may He be glorified, to grant you healing and
well-being, and to grant you relief from every hardship and a way out from
every difficulty. 

And Allah knows best.