It is never stated in the books, but Fudge has many qualities that could be attributed to any house.
But like this link(not canon link, but it shares my thoughts) says has all his qualities since they are hard to pinpoint into a house he can probably put into Hufflepuff. Because his qualities can never be pinpointed.
I would say that due to him never showing a lot of intelligence, I would not put him into Ravenclaw due to him never being extremely cunning.
He does try to meet his ends, but he also tries to help others. Like when he tried not to make everyone panic in the 4th and 5th book. That was partially from disbelief, but you could say that he is trying to be helpful and calming.
From here:
He is worried about the fallout of announcing Voldemort's return,
marking the end of the Wizarding world's years of peace, and the
sudden outbreak of gloom and terror; hence he decides to merely ignore
all of the evidence rather than accept the truth. J. K. Rowling has
since stated that Fudge's behavior mirrors that of Neville Chamberlain
in the lead-up to World War II
when he arrives at the school to take Hagrid to Azkaban, even though
he does not firmly believe that Hagrid is guilty. He also removes
Dumbledore as Headmaster when pressured by Lucius Malfoy who insisted
that all the school governors had voted on it.
He is impressionable, like his name suggests fudge. But that is also a common name(some what), but the quote still shows that he can be used. And that fact, not his power seeking, could be the reason he is in office.
But on another note, he could be a Slytherin.
He also passes a law allowing him to place Dolores Umbridge, his
Senior Undersecretary, as a teacher at Hogwarts. He then appoints
Umbridge as Hogwarts' "High Inquisitor", with the power to inspect and
sack teachers, and ultimately Dumbledore's successor as Headmaster,
which gives her (and by extension, Fudge himself) primary control of
how Hogwarts is managed. Fudge is concerned that Dumbledore is a
threat to his power and that he is planning to train the Hogwarts
students to overthrow the Ministry.
As from this quote he has a "addiction" to power, wanting to control peoples reactions, what Harry says and Hogwarts.
And in here he tries to hang onto his power.
He makes several attempts to have Dumbledore arrange a meeting between
himself and Harry so that Harry can lie on Fudge's behalf and make it
look as though the Ministry is winning the war, but Dumbledore
refuses, knowing how ridiculous Harry would find the idea. Fudge is
last mentioned in the series as one of the attendees at Dumbledore's
funeral; his fate during Voldemort's takeover of the Ministry during
the following year is unknown.
But he is a mix of both Slytherin and Hufflepuff.
"You are blinded by the love of your office(Note: Hufflepuff qualities), Cornelius! You place too
much importance, and you always have done, on the so-called purity of
blood!(Note: Slytherin qualities.) You fail to recognize that it matters not what someone is born,
but what they grow to be!"
—Albus Dumbledore to Cornelius Fudge
I would put him into Hufflepuff, due to him being not necessarily evil and the fact that he his mostly a good yet weak willed guy, but he his pushed toward wrong decisions by the fact that he has no leadership skills and he has the wrong people whispering in his ears.
From the wikia;
.... At the same time, Lucius Malfoy, then school governor, had
Dumbledore suspended, which Fudge tried to protest against, but in
futility. Eventually, it was proven that Hagrid was indeed innocent,
and he was released.
That summer, Sirius Black escaped from Azkaban, which unhinged Fudge
and started him on a downward spiral of poor decisions. A few weeks
prior to Black's escape, Fudge visited him while on a tour of Azkaban
prison and was unnerved at how normal Black seemed relative to the
other prisoners.
From the sorting hat songs:
Possibly he got his love of pureblood from being a Slytherin.
Said Slytherin, "We'll teach just those Whose ancestry's purest."
For Ravenclaw, the cleverest Would always be the best
Which can be seen from this quote below that Fudge is definitely not a Ravenclaw.
In Fudge's early days as Minister, it was said that he was not very
self-confident, and constantly sent letters to Albus Dumbledore,
asking for advice.
This is not canon, but it provides insight I don't agree with it(but it is insight).
But the songs can give you a idea of what it means to be in any house and can help you decide for yourself, as the songs give you a idea of what qualities are for each house.
I hope I answered you question , I will give a 1+ for making me think. But I could not find anything canon that said that answer outright.
