Mâliâraq Vebæk was born in Narsarmijit in the southernmost part of western Greenland. In 1932 she was sent to Aasiaat in northern Greenland, where the first continuation school for girls was set up that same year. She was subsequently sent with three other chosen girls to each live with a clergyman's family in Denmark in order to learn Danish.
In 1935 she was enrolled in the teacher training college Th. Langs Seminarium in Silkeborg to see how she fared: no woman from Greenland had ever before taken a higher education exam of this kind. In 1939 she returned to Greenland as a qualified primary school teacher, which meant that she came under Danish law and was thus entitled to a number of privileges such as free accommodation and fuel as well as eligibility to use the clergyman's farmhand to fetch water.
After the war, when she married the Danish archaeologist Leif Vebæk, she moved to Denmark, where she has lived since. She participated in her husband's field work in Greenland and gradually also began her own collection in the 1960s, recording a series of tales and songs in southern Greenland in the 1960s. Some of the collected material was published in 1983 in the book Niperujûht/Sydgrønlandske Sange.
She has also written poems, short stories, and radio plays and in 1981 published Búsime nâpíneq, the first Greenlandish novel written by a woman, translated into Danish a year later by the author herself and entitled Historien om Katrine. She subsequently published another novel, a number of short stories and translations, as well as Navaranaaq og andre, a history of Greenland women from the time of the legends to the present day. In 1995 she published the children's book Besøg hos Havets Moder.