I searched LibriVox, a site with free audiobooks of public domain books, for major Arthurian texts. I based my search terms off the Wikipedia page for the Matter of Britain, as well as the Bibliography of King Arthur. I cannot personally vouch for all of these, but I have listened to the History of the Kings of Britain, the Lays of Marie de France, and all three of the works by Chrétien de Troyes. (The Lays of Marie de France are my personal favorite, so far.) If you speak any languages besides English, you may want to search the site yourself to see if there are any Arthurian audiobooks read in that language, since I focused on English-language works or translations.

Arthurian Texts as Audiobooks

History of the Britons, Nennius, c.798. Translated by J.A. Giles, 1847.

History of the Kings of Britain, Geoffrey of Monmouth, c.1136. Translated by Aaron Thompson, 1718; translation revised by J.A. Giles, 1848.

Lays of Marie de France, Marie de France, c.1155-1170. Translated by Eugene Mason.

Erec and Enide, Chrétien de Troyes, c.1170. Translated by W.W. Comfort.

Lancelot, or The Knight of the Cart, Chrétien de Troyes, c.1177-1181. Translated by W.W. Comfort.

Yvain, or the Knight with the Lion, Chrétien de Troyes, c.1178-1181. Translated by W.W. Comfort.

The Mabinogion, anonymous, c.1350-1410. Translated by Charlotte Guest/Schreiber:
Volume 1
Volume 2
Volume 3

Gawain and the Green Knight, Gawain/Pearl Poet, late 14th century.
Translated by Jessie Laidlay Weston, c.1898. Version 1.
Translated by Jessie Laidlay Weston, c.1898. Version 2.
Translated by Charlton Miner Lewis, c.1903-1904. Version 1.
Translated by Charlton Miner Lewis, c.1903-1904. Version 2.
Translated by William Allan Neilson, c.1917.

Le Morte d'Arthur, Sir Thomas Mallory, c.1485:
Volume 1
Volume 2

The Birth of Merlin, or, The Child Hath Found his Father, William Rowley, 1552. Audio drama.

The Fairie Queene, Edmund Spenser, 1596. Version 1, multiple readers: Book 1, Book 2, Book 3, Book 4, Book 5, Books 6 and 7. Version 2, read by Thomas Copeland.

King Arthur, John Dryden, 1692. Audio drama.

"The Lady of Shalott", Alfred, Lord Tennyson, 1842. There are many readings of this poem available on LibriVox, so try this search for a full list. Two of these versions are in the An Arthurian Miscellany collection below.

Idylls of the King, Alfred, Lord Tennyson, 1856-1885.

Tristan and Isolde, Richard Wagner, c.1875. Translated by John P. Jackson, c.1885. Wagner's music drama, translated and performed as a spoken drama.

A Connecticut Yankee in King Arthur's Court, Mark Twain, 1889. Version 1, read by Steve Andersen. Version 2, read by John Greenman.

King Arthur, Joseph Comyns Carr, 1895. Audio drama.

Tristan and Iseult, Joseph Bedier, 1900. Translated by Hilaire Belloc, 1913.

Tristan and Iseult, Arthur Symons, 1917. Audio drama.

An Arthurian Miscellany, various (Alfred, Lord Tennyson, Algernon Charles Swinburne, Sara Teasdale, William Morris, and Thomas Bulfinch). Contents include: Alfred, Lord Tennyson's "Morte d'Arthur", "The Lady of Shalott" (1833 and 1843 versions), "Sir Launcelot and Queen Guinevere: A Fragment", and "Merlin and Vivien"; Algernon Charles Swinburne's "The Day Before the Trial"; Sara Teasdale's "Guenevere"; William Morris's "The Defence of Guenevere"; and Thomas Bulfinch's "The Round Table", Chapter 15 from The Age of Chivalry.