Vector notation was introduced to physics in the end of 19-th century. I am not 100% sure,
but this is associated in my memory with J. Willard Gibbs (Wikipedia confirms this, and also mentions Heaviside). 3-dimensional vectors were preceded by quaternions, and there was even a discussion what is better: vectors or quaternions.
Maxwell in his Treatease on Electricity and Magnetism (1873) did not use vectors, and
the standard mathematical textbook for physicists (Thomson and Tait, first edition 1867) does not mention them. Neither all multiple subsequent editions do.
As it is still not common knowledge, I will mention that vector notation is out of date.
It was superseded by the formalism of differential forms of E. Cartan
(invented in 1920-th). But differential forms still did not penetrate in undergraduate education, which seems very strange to me. And many (not all!) physicists continue to use
vector notation. Only a few of US universities teach differential forms to undergraduates.
EDIT. To those who do not believe what I wrote in the last paragraph, I recommend to look in the undergraduate textbook which is used in Harvard: Bamberg and Sternberg, A course in mathematics for students in physics, vol. 2, chapter Maxwell equations.
They explain by the way how notation evolved, from Maxwell's notation to
"vector calculus" to "tensor calculus" to modern language of differential forms.