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
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:
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,
2 Maccabees 2:4-8 states that the Prophet Jeremiah hid the Ark in a cave at Sinai along with some other consecrated items.
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.