No. Laozi did not write "When the student is ready, the teacher will appear." The sentence is not in the Tao Te Ching. Chapter 41 is the only passage in that book on students, and this saying is not that passage. The earliest close form is from 1885, in Mabel Collins, Light on the Path: "For when the disciple is ready the Master is ready also." The book belongs to Theosophical and occult literature. It is not a classical Chinese text that happens to be wearing English.
A smoother wording, in which "the teacher appears", comes later. It shows up in 1905, under Yogi Ramacharaka, a pen name of W. W. Atkinson. From there the sentence drifted onto Laozi, an easy name to hang a spiritual proverb on. Collins had said that the Master is ready also, not that a teacher materialises once the pupil feels prepared. Cite Light on the Path, 1885, if you cite a source. Leave the Tao Te Ching out of the note. Chapter 41 will not carry the line.