Are there any Rishonim who believe that God´s existence cannot be proven rationally?
וכן הוא מציאות ה' שהענין אמת אע"פי שאין לו מופת זו היא כונת בעל העקרים וראיתו יותר נכונה מכל ראיות הרב
And so it is [with regard to] the existence of God – it is true even though there is no proof for it. This is the intent of the author of The Ikkarim [R. Joseph Albo], and his evidence is better than all the evidence of the Rav [Rambam].
It depends what you mean by "rationally". There are rishonim who believed that a philosophical proof of G-d was either pointless or meaningless. For example, Rabbi Yehudah haLevi has the Chaver say in the first chapter of the Kuzari:
That which you are describing is religion based on speculation and system, the research of much thought, yet open to many doubts. Now ask the philosophers, and you will find that they do not agree on any one action nor any one principle, since some doctrines can be established by arguments, which are only partially satisfactory, and still much less capable of being proved. (Kuzari 1:13)
And then later:
There is an excuse for the Philosophers. Being Greeks, science and religion did not come to them as inheritances. They belong to the descendants of Yafes, who inhabited the north, while that knowledge coming from Adam, and supported by the divine influence, is only to be found among the progeny of Sheim, who represented the successors of Noah and constituted, as it were, his essence. This knowledge has always been connected with this essence, and will always remain so. The Greeks only received it when they became powerful, from Persia. The Persians had it from the Chaldaeans. It was only then that the famous [Greek] Philosophers arose, but as soon as Rome assumed political leadership they produced no philosopher worthy the name.
We have knowledge of G-d from trusted sources -- our own ancestors. The Greeks did the best they can with less inherited knowledge and much deduction, but as he said earlier, such results are unreliable and open to debate.
The Kuzari invokes something much like what epistemologists today call "Reliabilism". Which is a pretty rational reason to believe something. If you get a lot of useful advice from a medical textbook, at some point it's irrational to double-check every fact before using it. Similarly, the information from your parents or a tradition from our whole culture.
But it's likely not the kind of "rational proof" you were talking about.
It also draws a very different image of G-d. To the Rambam, one proves G-d Exists (Moreh Nevuchim, sec II, about the first third or so) and from there that such a G-d would talk to prophets, give a Torah, halakhah, etc... (Taking us through to the rest of the Moreh.)
To Rabbi Yehudah haLevi, we start similar to the way the Aseres haDiberos start, "I am Hashem your G-d Who took you out of the land of Egypt" (1:11):
I believe in the G-d of Avraham, Yitzchaq and Yisrael, the One Who led the children of Israel out of Egypt with signs and miracles; Who fed them in the desert and gave them a land, after having made them cross the [Red] Sea and the Jordan in a miraculous way; who sent Moshe with His Torah, and subsequently thousands of prophets, who confirmed His Law by promises to the observant, and threats to the disobedient. Our belief is comprised in the Torah--a very large domain.
The king of the Khazars is unhappy with this answer, expecting a more philosophical approach to G-d (par. 12):
I had not intended to ask any Jew, because I am aware of their low state and narrow-minded views, as their misery left them nothing admirable. Now shouldn't you, Jew, have said that you believe in the Creator of the world, its Governor and Guide, and in Him who created and sustains you, and such attributes which serve as evidence for every believer, and for the sake of which He pursues justice in order to resemble the Creator in His wisdom and justice?
And that is when the Chaver gives the response we saw above (par. 13), about how limited philosophy is at establishing truth. But his alternative is equally rationalist.
The Rambam's audience was influenced by a Moslem school of thought of the time called the Kalam, which eventually makes it to Christianity as Scholasticism. They take Aristotilian philosophy as discovered Truth, religion as revealed Truth, and try to make one picture of reality from the two. So his "Guide to the Perplexed" is a Jewish Qalam type picture.
The Kuzari was written before the Rambam's lifetime. (Perhaps it was a response to an even earlier Jewish Aristotilian, Rav Saadia Gaon and his "Emunos veDei'os".) However, the rishonim of Catalonia responded in the following centuries with a countervailing tendency. In particular, including the Ramban, the Ran, his student R' Yosef Albo and his Seifer haIqarim, and his student, Rav Chesdei Crescas in Or Hashem. There's was a staunchly rational approach, philosophical, but less accepting of Greek thought. Although in the language of the day, Or Hashem was an anti-philosophical work, that is only because "philosophy", then, was thought of as being the Greek kind in particular.
As per the name of his book, "Seifer haIqarim", Rav Albo believed that Judaism was founded on three postulates "iqarim" (essentials). From which one can derive 9 more mandatory beliefs, "shorashim" (roots), and many many other implications, "anafim" (branches). Interestingly, his list of 3 iqarim and 9 shorashim closely resembles the Rambam's 13 iqarei emunah (mandatory beliefs, none of which were held to be postulates). The 3 iqarim include belief that the universe has a Creator. (The other two are that the Creator revealed Truth to humans and that He metes out justice.) Belief in G-d is a postulate, not something proven from other postulates. (See Alex's answer.) Which is typical of this school.
Meanwhile, off in Ashkenaz, there was little organized study of philosophy. You can glean things about worldview between the cracks, by implications of comments made about Tanakh or halakhah. But nothing like the Ramban's commentary, where he would take the time to spell out a philosophical point.
An outlier case was one of the Tosafists, Rav Moshe ben Chasdai Taqu, who wrote a work called Kesav Tamim, of which only portions survived. He is usually assumed to be the one referred to by the Raavad when the Raavad objects to the Rambam labeling anyone who believes that Hashem has a body is a heretic. The Raavad says that at least one rabbi greater and more pious than the Rambam had such a belief.
However, the Kesav Tamim doesn't actually say that Hashem has a body, at least not in any text that survived. Rather, he so completely denies being able to reason about G-d that he says we cannot rule out Hashem having a body. In contrast to the Rambam, who says that G-d is so unfathomable all we can do is rule out what He isn't, Rav Moshe Taqu is saying all we can do is rely on prophecy and not reason at all.
A rational argument against rationalism. No?
That does not mean he insists we know for sure that phrases found in Tanakh like the "Hand of G-d" or "the anger of His Nostrils" was definitely meant as literal descriptions of G-d, as opposed to idioms referring to His Control and His Punishing the wicked. It seems more he was saying that in principle we cannot know for sure one way or the other. His primary topic is a polemic against R' Saadia Gaon and the Rambam for bringing philosophy to bear on the question, rather than trusting the Torah's words on their own terms.
So yes, there is at least one rishon who was anti-rationalist, saying that all theology is beyond reason. Then there were many rishonim who were anti-Kalam, who believed that the right way to ground religion was in something other than a philosophical proof from first principles. And yet, still held rational approaches.