The Ark of the Covenant is never recorded as having flown in Scripture.
There were only two means by the Ark ever moved -
Per God's command, the Ark was transported by the priests, using the Acacia poles which held it through the base, for this purpose.
When the Priests carried it into the Jordan River when it was still flowing in Joshua 5, for example, the river receded. If it could have flown, there would have been no need for the priests to "step out" in faith.
On two occasions, contrary to God's command, an ox cart was used to transport it.
a. In 2 Samuel 6, the Ark is being drawn "on a new ox cart," as follows:
And they carried the ark of God on a new cart and brought it out of the house of Abinadab, which was on the hill. And Uzzah and Ahio,[a] the sons of Abinadab, were driving the new cart,[b] 4 with the ark of God, and Ahio went before the ark.
Famously, it began to tip, and Uzziah tried to steady it. For this, he was immediately zapped (that's not a technical term) by God, and died instantly. Had the Ark been capable of flight, one would have assumed that it would have steadied itself in such an instance.
b. Also, in 1 Samuel 5, the Israelites had lost the Ark in battle to the Philistines. When the Philistine's local deity Dagon kept falling over before it, as if it were bowing, the Philistines started moving it from Ashdod to Gath and throughout their cities. Problem was- people kept dying. So, eventually they decided to get rid of it, and send it on its merry way with a bunch of gifts for the Israelites.
1 Samuel 6 records:
The men did so, and took two milk cows and yoked them to the cart and shut up their calves at home. 11 And they put the ark of the Lord on the cart and the box with the golden mice and the images of their tumors. 12 And the cows went straight in the direction of Beth-shemesh along one highway, lowing as they went. They turned neither to the right nor to the left, and the lords of the Philistines went after them as far as the border of Beth-shemesh. 13 Now the people of Beth-shemesh were reaping their wheat harvest in the valley. And when they lifted up their eyes and saw the ark, they rejoiced to see it. 14 The cart came into the field of Joshua of Beth-shemesh and stopped there. A great stone was there. And they split up the wood of the cart and offered the cows as a burnt offering to the Lord.
The story of the Ark of the Covenant being moved by Nazi
jeep pickup truck is, alas, non-canonical, and adheres only to the obscure Spielbergian sects of Christianity.
In all seriousness, however, the legends of the ark flying may be attributable to the winged cherubim on top of the mercy seat. This ornament of two angels, only either end of the ark, their wings touching, may have inspired such an idea, but they were strictly ornamental.