Valyrian weapons are unique weapons to the game which do not break but are difficult to acquire. They are also able to damage whitewalkers and wights.
1 or 2 handed Valyrian sword
A valyrian sword can be acquired from the weapons merchant in Qohor. But you will need a value of 3 or higher in town reputation for the option to buy it.
The merchant can make either two one handed or one two handed versions only. They cost 90,000 or 120,000 each respectively.
Enhancing the valyrian sword will result in a masterwork version which is visually different from the normal grade.
Pre-owned Valyrian Weapons
Number of lords of the game own a valyrian weapon. You can take it from them by either executing them or sending them to the wall.
Dawn is owned by Edryc Dayne. To get Dawn, you can get it by capturing him and sending him to the wall.
Long Claw is owned by Jon Snow. To get Longclaw, Jon must first be captured and be a prisoner. Upon rescuing him (ex: siege), there is a chance for him to join you. Then as a companion, you can take his weapon.
To obtain Ice you have to conquer Winterfell then look for the Weirwood tree, there will be a chest underneath the tree with Ice.
Item ID: 1036
Stannis has Lightbringer, you need to defeat him to get it
Blackfyre which is with Daenerys or Aegon, depending on wether you chose books or tv version (but always with the Targaryen invader).
Dark Sister is in the mod but idk if Produno already made it available, either way it's obtainable by quest if and when it is added.
Paxter Redwyne has Redwyne.
Randall Tarly has Heartsbane
Item ID: 1039