Scatha the Worm

In the appendices of the Lord of the Rings we are told about Scatha the Worm who was slain by Fram of Rohan.

‘Many lords and warriors, and many fair and valiant women, are named in the songs of Rohan that still remember the North. Frumgar, they say, was the name of the chieftain who led his people to Éothéod. Of his son, Fram, they tell that he slew Scatha, the great dragon of Ered Mithrin, and the land had peace from the long-worms afterwards. Thus Fram won great wealth, but was at feud with the Dwarves, who claimed the hoard of Scatha. Fram would not yield them a penny, and sent to them instead the teeth of Scatha made into a necklace, saying: “Jewels such as these you will not match in your treasuries, for they are hard to come by.” Some say that the Dwarves slew Fram for this insult. There was no great love between Éothéod and the Dwarves.
_{The Lord of the Rings - Appendix A - The House of Eorl}

This is also mentioned in the main text.

Then Éowyn gave to Merry an ancient horn, small but cunningly wrought all of fair silver with a baldric of green; and wrights had engraven upon it swift horsemen riding in a line that wound about it from the tip to the mouth; and there were set runes of great virtue.

‘This is an heirloom of our house,’ said Éowyn. ‘It was made by the Dwarves, and came from the hoard of Scatha the Worm. Eorl the Young brought it from the North. He that blows it at need shall set fear in the hearts of his enemies and joy in the hearts of his friends, and they shall hear him and come to him.’ _{The Lord of the Rings - Book 6, Chapter 6 - "Many Partings"}

However it should be noted that this was all prior to when Rohan settled the land that we see them in in The Lord of the Rings. So although Scatha was attacking the lands that Rohan was living in at the time, it was different than their later lands.

There are no recorded dragon attacks in any of Tolkien's writings on the geographic lands where Rohan and Gondor would eventually settle.

This is not to necessarily say that it never happened, but one would think that if it was after the Numenoreans settled there in the middle of the second age Tolkien would have mentioned in the appendices, and it also doesn't seem like Morgoth was sending his dragons that far afield in the first age. Such an attack could have conceivably happened in the early years of the second age, but if it did Tolkien never wrote about it.

Scatha the Worm

Related to this, we do know of a dragon that attacked the people of Rohan prior to when they had settled the land that we see them in in The Lord of the Rings.

This is Scatha the Worm who was slain by Fram of Rohan. We learn his story in the appendices of the Lord of the Rings.

‘Many lords and warriors, and many fair and valiant women, are named in the songs of Rohan that still remember the North. Frumgar, they say, was the name of the chieftain who led his people to Éothéod. Of his son, Fram, they tell that he slew Scatha, the great dragon of Ered Mithrin, and the land had peace from the long-worms afterwards. Thus Fram won great wealth, but was at feud with the Dwarves, who claimed the hoard of Scatha. Fram would not yield them a penny, and sent to them instead the teeth of Scatha made into a necklace, saying: “Jewels such as these you will not match in your treasuries, for they are hard to come by.” Some say that the Dwarves slew Fram for this insult. There was no great love between Éothéod and the Dwarves.
_{The Lord of the Rings - Appendix A - The House of Eorl}

This is also mentioned in the main text.

Then Éowyn gave to Merry an ancient horn, small but cunningly wrought all of fair silver with a baldric of green; and wrights had engraven upon it swift horsemen riding in a line that wound about it from the tip to the mouth; and there were set runes of great virtue.

‘This is an heirloom of our house,’ said Éowyn. ‘It was made by the Dwarves, and came from the hoard of Scatha the Worm. Eorl the Young brought it from the North. He that blows it at need shall set fear in the hearts of his enemies and joy in the hearts of his friends, and they shall hear him and come to him.’ 
_{The Lord of the Rings - Book 6, Chapter 6 - "Many Partings"}