Were Jews allowed to sacrifice to the LORD at altars other than at the tabernacle and temple?

After King Solomon, The kingdom split in two due to varies disagreements and abuses started with Solomon and Continuing with his Son Rehoboam. As a result, 10 tribes split from the temple worship in Jerusalem and started the Northern Kingdom of Israel; the Southern Kingdom was called the Kingdom of Judah. Judah is where we get the word "Jew”.

When Jeroboam left and separated from Judah, the Northern Kingdom developed their own ways of worshiping and offering sacrifice, they also built two temples so that they could separate themselves from Jerusalem. Very sadly, many of those in the Northern Kingdom turned to idolatry and other practices.

The result of these alternate practices included altars made to God that were different and separate from the sacrificial worship in the temple.

Moving forward in time, after the Babylonian and Assyrian exiles, the ten tribes of the North, having returned to the land, were seen by the Kingdom of Jerusalem as unclean, having mixed there blood with the Babylonians. Having come from a land whose capital was Samaria, They were called Samaritans. Now, the Samaritans referred to themselves as Israelites, having come from the Kingdom of Israel.

What came from the separation of the 12 tribes is separate ways of worshiping. One, which comes form Judah, and one that comes from Israel. The two separate Israelite communities exist even today.

The answer to temples worship outside of the Jerusalem temple may be as simple as those temples being temples of the Kingdom of Judah or Samaritan temples, before and during and after the Babylonian and Assyrian exiles.

To answer the question. Were Jews allowed to offer sacrifice outside the Temple, the answer would be "No" - not those sacrifices listed in Leviticus. The Problem was the separation of the tribes, which did not go well for either the northern or southern kingdoms. Some of the 12 tribes, those separated from Judah, and in the Kingdom of Israel, did offer sacrifice to God outside the Temple. The Judahites, aka "jews" did not.

While Yahwistic temples and worship sites outside Jerusalem may appear to violate the Deuturonomic prohibition, this is only so because we often think of the history of ancient Israel and Judah backwards. The broad consensus of scholars is that the Book of Deuteronomy – though set by its authors in an earlier era – was actually written several centuries after the period of the judges and the kings. That is, the outlying worship sites were not prohibited during the time of their actual use. The prohibition only arose when the emerging monotheistic religion was later centralized. As Ephraim Stern notes:

"We sometimes get the impression that after Solomon built the Temple in Jerusalem, Yahweh had no other sanctuary in ancient Israel—but this is not the case. The religious reforms of, first, King Hezekiah in the late eighth century B.C.E. (2 Kings 18:1–8; 2 Chronicles 29–31), and then of King Josiah, in the late seventh century B.C.E. (2 Chronicles 34–35), sought to centralize the worship of Yahweh in Jerusalem. The need for these reforms demonstrates, however, that Yahweh worship was by no means confined to Jerusalem."

The Deuteronomic History that told stories about the older sites is believed to have been written centuries after Elijah. These stories do not contradict the Law so much as they contain remnants of history that predate the Law itself. Understanding how the religion of Yahweh and the biblical texts actually developed and achieved their mature form within early Judaism during and after the Exile – and how these differ from the story told within the Bible itself – is critical to resolving exactly these kinds of puzzles in the text. It is only when we attempt to enforce a literary timeline on real events that 'contradictions' appear.

Many places in the Old Testament show the people of Israel either being directed by God, or of their own accord, building altars to Him that were not either in the tabernacle, or (much later) the temple.

A sampling of passages:

• Deuteronomy 27:4ff, "So when you have crossed over the Jordan, you shall set up these stones, about which I am commanding you today, on Mount Ebal, and you shall cover them with plaster. And you shall build an altar there to the Lord your God, an altar of stones on which you have not used an iron tool. You must build the altar of the Lord your God of unhewn stones. Then offer up burnt offerings on it to the Lord your God, make sacrifices of well-being, and eat them there, rejoicing before the Lord your God. You shall write on the stones all the words of this law very clearly." (followed up in Joshua 8:30)
• Joshua 22:10, "When they came to the region[a] near the Jordan that lies in the land of Canaan, the Reubenites and the Gadites and the half-tribe of Manasseh built there an altar by the Jordan, an altar of great size."
• Judges 6:24a, "Then Gideon built an altar there to the Lord, and called it, The Lord is peace."
• Judges 13:19-20, "So Manoah took the kid with the grain offering, and offered it on the rock to the Lord, to him who works wonders. When the flame went up toward heaven from the altar, the angel of the Lord ascended in the flame of the altar while Manoah and his wife looked on; and they fell on their faces to the ground."
• Judges 21:4, "On the next day, the people got up early, and built an altar there, and offered burnt offerings and sacrifices of well-being."
• 1 Samuel 14:35, "And Saul built an altar to the Lord; it was the first altar that he built to the Lord."
• 2 Samuel 24:18, "That day Gad came to David and said to him, “Go up and erect an altar to the Lord on the threshing floor of Araunah the Jebusite.”" (followed up later in the chapter)
• 1 Kings 18:30, "Then Elijah said to all the people, “Come closer to me”; and all the people came closer to him. First he repaired the altar of the Lord that had been thrown down;"

Worship offered at those altars appears to not be the same as the Levitical worship prescribed to the people of Israel - ie, they were meant for specific purposes at specific times, and not for "general" or "normal" worship.