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"
(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.