As the title asks, does the Bible indicate when, where, or how the ark of the covenant finally disappeared?
If not, do we have any likely clues?
No, the Bible does not. This is the last mention of the ark in the Old Testament:
2 Chronicles 35:3
And said unto the Levites that taught all Israel, which were holy unto the LORD, Put the holy ark in the house which Solomon the son of David king of Israel did build; it shall not be a burden upon your shoulders: serve now the LORD your God, and his people Israel,
After that, it fades from sight for the rest of the Bible until the apostle John sees a vision of an ark in Heaven in the last book of the Bible:
And the temple of God was opened in heaven, and there was seen in his temple the ark of his testament: and there were lightnings, and voices, and thunderings, and an earthquake, and great hail.
How it got there, or if it is the same ark, we don't know.
Some Christian traditions include the Book of Maccabees in their Biblical canon. From Wikipedia,
The Books of the Maccabees are books concerned with the Maccabees, the leaders of the Jewish rebellion against the Seleucid dynasty, or related subjects.
The term mostly refers to two deuterocanonical books contained in some canons of the Bible:
1 Maccabees, originally written in Hebrew and surviving in a Greek translation, relates the history of the Maccabees from 175 BC until 134 BC.
2 Maccabees, a Greek abridgment of an earlier history in Hebrew, relating the history of the Maccabees down to 161 BC, focusing on Judas Maccabaeus, talks about praying for the dead, offerings.
2 Maccabees 2:4-8 states that the Prophet Jeremiah hid the Ark in a cave at Sinai along with some other consecrated items.
It was also in the same document that the prophet, having received an oracle, ordered that the tent and the ark should follow with him, and that he went out to the mountain where Moses had gone up and had seen the inheritance of God. Jeremiah came and found a cave-dwelling, and he brought there the tent and the ark and the altar of incense; then he sealed up the entrance. Some of those who followed him came up intending to mark the way, but could not find it. When Jeremiah learned of it, he rebuked them and declared: ‘The place shall remain unknown until God gathers his people together again and shows his mercy. Then the Lord will disclose these things, and the glory of the Lord and the cloud will appear, as they were shown in the case of Moses, and as Solomon asked that the place should be specially consecrated.’
According to modern scholars' chronology of the ministry of Jeremiah, this dates the disappearance of the Ark to somewhere around the early 6th century BCE.
I think you'll find the story in the Kebra Negast, rather than the Bible.
See Littman, Dr E, The Legend of the Queen of Sheba in the Tradition of Axum, Book 1 of Biliotheca Abessinica: Studies Concerning the Languages, Literature and History of Abyssinia, Princeton, Leyden and The University Library: E J Brill, 1904, p 11.
As the story goes, King Solomon and the Queen of Sheba had a son, who stole one of two Arks from Solomon and took it back to Ethiopia, where, according to some sources, it remains to this day.
As for when the alleged transfer might have happened, the accepted dates for King Solomon’s reign are between 970 and 931 BC. Solomon is supposed to have been alive at the time of the theft, so that should give you a rough time frame.
According to this, there is tradition that when Queen Sheeba son's Menelik I was of 19 years old, he wanted to see his dad (King Solomon) and spend sometime with him. After staying with him for some years, he wanted to have a replica of Ark to take with him. So Solomon made him a replica, which he was to take with him to Ethiopia. When he was leaving, he replaced the replica with the original as he thought it was not secure in Israel. Today, there are "Keepers of Ark" who take care of the building and protect it.
The place where they kept the Ark in Ethiopia was recently in the news.