Can I marry a Quranist who does not believe in the Prophet's Sunnah?

This belief contradicts the fundamentals of the Islamic doctrine. You need to advise this young man to seek sound teachings of Islam from authentic prominent Muslim scholars. You must not accept his marriage proposal until he adheres to the methodology of the Sunnis with respect to such fundamentals of the faith.

The prophetic traditions (the Sunnah) are the second legal source for Islamic legislation and serves as supplementary to the Quranic text which forms the first legal source of Shari’ah. Scholars opined that the prophetic traditions play different roles as it can be explanatory to the Quranic text and it also can be independent source for legislation.

Definition: The sunnah is defined as “statements, actions, tacit approvals, or qualities related about the Prophet (upon him be peace).”

In terms of statements, we may take the example of that which is related from Umar b. Al-Khattab, who said: “I heard the Prophet (peace be upon him) saying, ‘Actions are only by intentions. So, every man will have that which he intended.”

In the category of actions are included all actions of the Prophet that have reached us, such as his manner of doing ablutions (wudu’), and of performing the pilgrimage (hajj). And so, we have a hadith from the Prophet, saying “Pray as you have seen me pray.” Similarly in his final pilgrimage, he said, “Take the hajj rituals from me”.

Tacit approvals consist of the Prophet’s affirmations of the statements or actions of the Companions, either through his silence (and, therefore, lack of repudiation), or through his express agreement and commendation. Examples of this include the narration of Abu Sa’id al-Khudri who said, “Two men set out on travel. When the time for prayer came, neither of them had any water with him. So they performed tayammum and prayed. Thereafter, they came across water while the time of prayer had not expired. One of them performed his regular ablutions and repeated his prayer, while the other did not.

Then, they came to the Prophet (peace be upon him) and mentioned to him what had transpired. He said to the one who did not repeat the prayer, ‘You have achieved the sunna, and your prayer was valid.’ And to the one who performed the ablutions and prayed again, he said, ‘You have two rewards.’”

As for qualities, many of the Prophet’s qualities and characteristic have been related. Some scholars have even taken to compiling these, such as al-Tirmidhi in his work, al-Shama’il
The types of rulings found in the sunna:

1. Rulings according with and emphasizing the rulings of the Quran. An example of this is a hadith related by Anas b. Malik who said that the Prophet (peace be upon him) said, “The wealth of a Muslim is not permissible to another except with his agreement and permission.” This is in agreement with the verse, “O ye who believe! Squander not your wealth among yourselves in vanity, except it be a trade by mutual consent,” (al-Nisa: 29). 2. Rulings that clarify and explain general commandments in the Qur’an. An example of this is that which clarifies the amounts for zakat. 3. Rulings that restrict or specify general commands of the Qur’an. An example is the general Qur’anic command to amputate the hand of the thief, which the sunna specified is to be done at the wrist. Another example is the general Qur’anic prohibition against eating carrion: “Forbidden to you (for food) are: dead meat, blood, the flesh of swine, and that on which hath been invoked the name of other than Allah.” (al-Ma’ida: 3). However, the following hadith from Ibn ‘Umar exempts seafood from this prohibition. According to Ibn ‘Umar, the Prophet said, “Two types of blood and two types of dead meat have been made permissible for us: of the dead meat, fish and locusts; and of the blood, liver and spleen.”

4. New rulings not mentioned in the Qur’an, because the sunna is an independent source of legislation. In this regard, it is not unlike the Qur’an. It is reliably transmitted that the Prophet said, “I have been given the Book and something like it with it. I have been given the Qur’an and something like it with it.... So mandatory upon you is the Qur’an. What you find deemed permissible in it has been made permissible for you, and what what you find deemed impermissible in it has been made impermissible for you.

So, neither the domesticated donkey nor any predator with fangs is permissible for you.” This means the Prophet was given the Qur’an and something like it, i.e. the sunna. The rulings of both must be followed, and from this we come to know of the prohibition against eating both domesticated donkeys and fanged predators, as well as the prohibition against eating birds with claws, and the rulings on the inheritance of the grandmother, etc.Therefore, denying the Prophet’s sunnah tantamount to denying a fundamental base of Islamic legislation.