About marriage and its necessity for both having intercourse and pregnancy
In Islam it is necessary to marry in order to have intercourse and have children.
And it is sunnah because it is the practice and recommendation of our prophet ():
... "O young men, those of you who can support a wife should marry, for it (marriage) controls the gaze and preserves one from immorality. (See for example in Bulugh al-Maram)
Imam al-Bahuti البهوتي said about "nikah" (marriage in Islam) in his ar-Rawd al-Murabba' the commentary on Zaad al-Mustanqi' الروض المربع شرح زاد المستقنع -see here-:
In the following I'll translate from Arabic language,a s the translation is of my own take it with the necessary care!
Further note that the terms are defined from the male perspective!
(وهو سنة) لذي شهوة لا يخاف زنا من رجل وامرأة
(It is sunnah) for those who have desires and don't worry for committing zina both men or women.
...
ويباح لمن لا شهوة له كالعنين والكبير،
and it is permissible for those who have no desires like the old or impotent.(وفعله مع الشهوة أفضل من نوافل العبادة)
(And performing it due to desire is better than nafl worship)لاشتماله على مصالح كثيرة، كتحصين فرجه وفرج زوجته، والقيام بها، وتحصيل النسل وتكثير الأمة، وتحقيق مباهاة النبي - صَلَّى اللَّهُ عَلَيْهِ وَسَلَّمَ - وغير ذلك،
This is because it includes many benefits like guarding the private parts of oneself and the wife, and being in charge for her, and having offspring, and increase the great number of the ummah, and realizing the boast of the prophet -Allah's praying and blessings be among him- and other reasons.ومن لا شهوة له نوافل العبادة أفضل له.
As for those who have no (such) desires the nafl worship is better.
Rights (and duties) of spouses after marriage
But you asked about the necessity of intercourse or the procreation of offspring:
Anything else than above is a matter that the to be spouses should agree upon in best case before getting married. See for example fatwa islamweb #156219, which shows that it is basically permissible to get married and not have intercourse, when mutual agreement is given.
But If any of both wants children it is his/her right to have them (be it with the spouse after agreement or by a new marriage).
Nevertheless you should be aware that you'll hardly find a (young) husband or even a (young) wife that would reject having intercourse at all.
Therefore expecting a sane (young) person to accept or agree about not having intercourse after marriage is rather an unworldly illusion.
A bit off-topic: Mahr only becomes obligatory if marriage was consumed
Especially as the consensus is that mahr is only a due on the husband after consumption of the marriage (by intercourse) based on (See also: After divorce followed by converting back to Christianity, does an ex-wife need to return the meher?):
There is no blame upon you if you divorce women you have not touched nor specified for them an obligation. But give them [a gift of] compensation - the wealthy according to his capability and the poor according to his capability - a provision according to what is acceptable, a duty upon the doers of good. (2:236)
and the fact that:
... And live with them in kindness. For if you dislike them - perhaps you dislike a thing and Allah makes therein much good. (4:19)
implies beside the right of treating wives the right way (in kindness) and behaving in a manner she likes indirectly as al-Hassan al-Basri -as quoted by al-Bagahwi in his tafssir- added the right of treating them well in bed or at least in what is related to physical activities between spouses like hugs and kisses etc..
Spouses are allowed to find mutual agreements one could also remain unmarried
So maybe if you find an understanding husband you may both find a solution or arrangement and help for your situation that may satisfy your needs and help you come over your phobia.
Else you should seek professional help to learn to control, over come or "fight" your phobia.
We don't know the reasons maybe it is something that was addressed in the fatwa from islamqa linked below!
Or you may remain unmarried as this certainly is permissible. Even if many scholars hold the opinion that it is wajib if one is able to afford (age etc.) it.
Note that in worst case talaq is an option too.
See also these relevant posts:
Can a women take tablets not to get pregnant
Does the wife have the right to reject sex?
Is it okay for a Muslim girl not to have children?
Wife's Tamkin and Marital Rape
and this fatwa on islamqa #241951 which rather looks like an consultation.