“When the student is ready the teacher will appear”: did Laozi say it?
Not in the Dao De Jing
No: “When the student is ready the teacher will appear” is not in the Dao De Jing; it grew out of Theosophy and Western occultism around 1900, and Laozi's name was added a century later.
When the student is ready the teacher will appear. When the student is truly ready the teacher will disappear.
The nearest lines in the Dao De Jing
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故善人者，不善人之师；
Hence, the benefactor is the teacher of the lesser person,
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不善人者，善人之资。
the lesser person is the endowment of the benefactor.
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不贵其师，不爱其资，
Fail to value the teacher, fail to like the endowment,
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上士闻道，
When the superior scholar hears of the Way,
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勤而行之；
he practices it diligently;
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中士闻道，
when the average scholar hears of the Way,
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若存若亡；
he half keeps it, half loses it;
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下士闻道，大笑之。
when the lesser scholar hears of the Way, he laughs out loud at it.
Where it comes from
The saying is in no chapter of the Dao De Jing. Stefan Stenudd, who keeps a list of sayings wrongly credited to Laozi, follows its first half back through Western occultism: Mabel Collins's Light on the Path (1885) has “when the disciple is ready the Master is ready also”, a Theosophical magazine of 1914 has “When the pupil is ready, the Master will appear”, and William Walker Atkinson, writing as Yogi Ramacharaka in 1905, has “when the student is ready the teacher appears”. The second half, the teacher who disappears, he finds first in a 1995 book by the psychologist Frances Vaughan. The whole saying is credited to Laozi online from 2014 and in books from 2016, with no source.
What the Dao De Jing says instead
The book speaks of a teacher in chapter 27, and not of readiness: 不善人之师, “Hence, the benefactor is the teacher of the lesser person, the lesser person is the endowment of the benefactor”: each needs the other. Chapter 41 ranks those who hear of the Way: the superior scholar “practices it diligently”, the lesser one “laughs out loud at it.” Readiness there lies in the hearer, and no teacher appears or leaves.