Sylvia Plath mentioned sleeping pills in numerous letters. The pills were barbiturates, some available over-the-counter.
From Sylvia Plath's Letters, Volume 1:
28 June 1949 to Aurelia Plath: "I really don't need any sleeping pills by the time the day is over." (Plath at the time is 16 years old.)
29 October 1950 to Aurelia: "I have taken a hot bath and a sleeping pill"
3 December 1950 to Aurelia: "I'm going to take two little pills and be asleep by nine o'clock"
31 January 1951 to Aurelia: "I guess I'll take sleeping pills till after exams are over"
9 May 1951 to Ann Davidow: "swallowing a handful of cold pills"
18 October 1951 to Aurelia: "It's wonderful how comfortable strong nosedrops, hot compresses and penicillin and sleeping pills can make a sinus sufferer."
20 October 1951 to Aurelia: "I was dosed with pirivine [nose drops] and pyribenzamine [antihistamine] and sleeping pills"
3 March 1952 to Aurelia: "I took two sleeping pills two hours apart, as you said"
17 November 1952 to Aurelia: "got the first good night's sleep (with the aid of a phenobarb) for weeks."
(Plath takes sleeping pills prescribed by her aunt and family doctor, Francesca Racioppi, M.D., for most of spring semester 1953. That summer, Racioppi prescribes for Plath a stronger sleeping pill. In August, Plath overdoses on sleeping pills and barely survives.)
28 December 1953 to Eddie Cohen: "I became immune to increased doses of sleeping pills"
27 October 1955 to Elinor Friedman: "Gone are the good old Smith days with cocaine, codeine, and sleeping pills" [1]
Journals, 5 November 1957: "First I couldn't sleep without pills, now I can"
In Letters, Volume 2, as Sylvia's marriage falls apart she becomes addicted to sleeping pills and knows it.
29 September 1962 to Kathy Kane: "I can't sleep without pills."
12 October 1962 to Aurelia: "Every morning when my sleeping pills wear off, I am up about 5, in my study with coffee, writing like mad."
16 October 1962 to Aurelia: "I live on sleeping pills"
14 December 1962, to Dorothy Benotti: "I hope to get off sleeping pills"
26 December 1962, to Daniel and Helga Huws: "I am going to the doctor this week to see if he can help me get off those sleeping pills"
4 February 1963, to Ruth Beuscher: "I am living on sleeping pills and nerve tonic"
Because most of Sylvia's mentions of sleeping pills are in letters to her mother, I think her mother, who herself took sleeping pills, got her started on them. There are very few studies about Plath and drugs.
[1] U.S. pharmacies legally sold cocaine and the morphine derivative codeine most of the 20th century, at first over-the-counter and then by prescription. Sylvia Plath was prescribed cocaine for sinus pain.