Brill lists three translations:
Eitel, E.J. “Mu-t’ien-tzu chuan.” China Review 17 (1888): 223–40; 247–58.
Cheng Te-k’un. “The Travels of Emperor Mu.” JNCBRAS 64 (1933): 124–42; 65(1934): 128–49.
Mathieu, Rémi. Le Mu tianzi zhuan, traduction annotée, etude critique. Paris: Collège de France, Institut des Hautes Etudes Chinoises, 1978.
Presumably with the first two being English and the third being French.
E.J. Eitel's translation entitled Mu T'ien-tzu Chuan (Muh-T'ien-Tsze Chuen) -or- The Chronicle of Mu, Son-of Heaven was split into two parts and both are hosted on the Hong Kong Journals Online section of the Hong Kong University website.
As for the Cheng Te-k’un translation the best I've got so far is found in the Classical Chinese Literature: From antiquity to the Tang dynasty
edited by John Minford, Joseph S. M. Lau
The Wheels of His Chariot, The Hoofs of His Horses Selections from The Travels of Mu, Son of Heaven
Translated by Cheng Te-k'un
Starts on page 59:
On the day mou-yin the emperor made his way to the west, marching on as far as the Yangyu Mountain, where in ancient days Wu Yi, the God of the River, had established his family, the house of Hezong. A member of this house, Bo Yao, welcomed the emperor at the Yanren Mountain, offering as presents a piece of silk ...
It's only selections but they say the full length is:
This also matches the, more specific, Brill tag. It seems to be split between journals, much like the Eitel translation, and it should be found in:
Journal of the North China Branch of the Royal Asiatic Society
Journal of the North China Branch of the Royal Asiatic Society (皇家亚洲文会北华支会会刊) has combined and "photocopied" (read:re-printed(?)) all 75 volumes of their journals and you can get it on Amazon for ￥25,900.00.
But really you're only looking for the 33rd volume in the new printings because it contains:
Volumes 64 and 65 of the original journals.