There's a possibility that it existed, even if it wasn't made by Moses. When the temple in Jerusalem was restored (See 2Kings 22) by priest Hilkiah, the book of the Law was discovered. The scholar consensus (I can mention Israel Finkelstein and William G. Dever as two archaeologists that support this idea) is that the book of the Law was not discovered, but written by priest Hilkiah himself. Most probably this book was the book of Deuteronomy. And Deuteronomy says:

"Take this Book of Instruction and place it beside the Ark of the Covenant of the LORD your God, so it may remain there as a witness against the people of Israel." (Deut 31:26)

So, if the book of Deuteronomy was an invention dated to the 7th century BCE, it would be illogical to think that the priests wrote that passage without thinking of making an ark referenced in that passage.

Another author, Richard Elliott Friedman (who wrote the Bible?) says that there was some rivalry between Judah and Israel, and some passages in the Torah were written to legitimize Judah over Israel, like the passage of the golden calf. Friedman also explains (forgot the page, sorry) that the rivalry began because the cult in Israel did not require to have an ark for worship - instead, they made a golden Calf, as told in the book of kings:

After seeking advice, the king made two golden calves. He said to the people, “It is too much for you to go up to Jerusalem. Here are your gods, Israel, who brought you up out of Egypt.” One he set up in Bethel, and the other in Dan. 30 And this thing became a sin; the people came to worship the one at Bethel and went as far as Dan to worship the other.

(1Kings 12:25-33)

So, according to Friedman, the famous passage about the golden Calf that Aaron made in Exodus 32 was supposed to be a retcon to condemn Jeroboam and the worship of the golden calves in Dan and Bethel. Later Friedman digs a little deeper into the story and explains how there was a lot of rivalry between distinct priests who fought for political power.

Then again, these are just hypotheses. If the ark existed, it's most probable that it was burnt by the Assyrians to extract the gold plating from it. So we'll never know.

But at least there's a chance that the Ark did in fact exist.

It is actually possible that the Ark of the Covenant still exists.

The Ethiopians have long claimed that the Ark of the Covenant was moved to Ethiopia 3,000 years ago and still resides there in a chapel in the town of Aksum, protected by local monks.

http://www.smithsonianmag.com/people-places/keepers-of-the-lost-ark-179998820/

Whether this claim is true or not is hard to prove, since the monks will not let anyone investigate the arc. It was declared to be true, however, by Abuna Paulos, the Patriarch of the Ethiopian Orthodox Church (PhD from Princeton). He did state, however, that even he was not allowed to see the Ark. Only one monk, who is the guardian of the Ark really knows what it looks like.