Were only priests allowed to offer sacrifices?

Answer:

The judges I could find who offered sacrifices were:

Gideon - "Now it came to pass the same night that the LORD said to him, "Take your father's young bull, the second bull of seven years old, and tear down the altar of Baal that your father has, and cut down the wooden image that is beside it; and build an altar to the LORD your God on top of this rock in the proper arrangement, and take the second bull and offer a burnt sacrifice with the wood of the image which you shall cut down." So Gideon took ten men from among his servants and did as the LORD had said to him. But because he feared his father's household and the men of the city too much to do it by day, he did it by night" (Judges 6:25-27).

Eli - Implied because he and his sons were priests of God (I Samuel 1:3; 2:11).

Samuel - "So Samuel did what the LORD said, and went to Bethlehem. And the elders of the town trembled at his coming, and said, "Do you come peaceably?" And he said, "Peaceably; I have come to sacrifice to the LORD. Sanctify yourselves, and come with me to the sacrifice." Then he consecrated Jesse and his sons and invited them to the sacrifice" (I Samuel 16:4-5).

Gideon was of the western half of the tribe of Manasseh (Judges 6:11; Joshua 17:2). While he was not a priest or even of the tribe of Levi, he was given a direct order by God to offer the sacrifice. Thus, as a prophet of God, he was permitted to do so.

Eli was a priest and therefore required to give sacrifices.

Samuel was a Levite from Ephraim (I Samuel 1:1; I Chronicles 6:33-38). All priests are Levites, but not all Levites are priests. However, Samuel was the firstborn of his mother and was dedicated to the Lord through her vow (I Samuel 1:11). There were special rules concerning the firstborn. They were dedicated to God by law (Exodus 13:2, 12-15). Originally the firstborns were to be the priests of God, but they were later replaced by the Levites. Parents were required to redeem them from God in acknowledgment of the Levites being dedicated to God. Children vowed to God could also be redeemed (Leviticus 27), but something redeemed once could not be redeemed a second time. By vowing her firstborn son to God, Hannah had dedicated him for life to God. In essence, Samuel was adapted in as a priest under the law that existed prior to the Levites becoming servants of God. (The article "Jephthah's Daughter" has more on these rules. See also "How could Samuel offer a burnt offering when he was not a Levite?")

In addition, Samuel was a prophet of God and could offer sacrifices at the order of God. This is how Elijah, of the tribe of Gad, had the right to offer his famous sacrifice on Mount Carmel.

It is not so much that only the priests could offer sacrifices. The point has always been that God selects who may offer sacrifices to Him. He appointed the descendants of Aaron to be His priests, thus their authority came from God. God told others to offer sacrifices, so they too operated under the order of God. Saul, however, decided on his own to make an offering to God and as a result, was condemned for it. "And Samuel said to Saul, "You have done foolishly. You have not kept the commandment of the LORD your God, which He commanded you. For now the LORD would have established your kingdom over Israel forever" (I Samuel 13:13; see I Samuel 10:8 for the original command).