One of the earliest attempts to develop a calculus for working directly on vectors was made by Gottfried Wilhelm von Leibniz in $1679$, but this was unsuccessful.
Jean-Robert Argand's demonstration in $1806$ of a geometrical representation of the complex plane gave the misleading impression that vectors in the Cartesian plane required them to be represented as complex numbers, which held development back for some time.
August Ferdinand Möbius published his Der Barycentrische Calcul in $1827$, which was the forerunner of the more general analysis of geometric forms developed by Hermann Günter Grassmann.
Giusto Bellavitis published Calcolo delle Equipollenze in $1832$, which was one of the first works to deal systematically with addition and equality of vectors.
In $1843$, Hermann Günter Grassmann published Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (that is: "Linear Extension Theory, a new branch of mathematics").
In $1844$, William Rowan Hamilton started publication of a series of articles in Philosophical Magazine discussing quaternions.
Both of these works developed the theory of vector analysis, independently of each other, from different directions.
Further development was due to Peter Guthrie Tait, whose Elementary Treatise on Quaternions of $1867$ progressed the theory considerably.
However, the theory of quaternions was too complicated and theoretical to be much practical use in studying real-world problems.
As a result, several mathematical physicists worked on improving the system and developing more elementary techniques.
Important to this process were August Otto Föppl, Max Abraham, Alfred Heinrich Bucherer, Vladimir Sergeyevitch Ignatowski, Richard Martin Gans, Oliver Heaviside and Josiah Willard Gibbs.
The approach of Gibbs and Heaviside was not well received by Tait, who was displeased with the fact that they did not use his beloved quaternions.
Contrariwise, Gibbs and Heaviside did not appreciate the inflexibility of Tait's approach, being likened by them to (and ridiculed as) a religious ritual.
However, the approach of Gibbs and Heaviside prevailed, and by the time of Edwin Bidwell Wilson, their techniques had bedded in.
C.E. Weatherburn gleefully relates the controversy, standing four-square upon the side of Gibbs and Heaviside, as well he might; his presentation is thoroughly within their tradition.
Roberto Marcolongo and Cesare Burali-Forti continued the work in developing vector algebra.
It is worth noting that many of the techniques of vector analysis were developed in response to the need to analyse Maxwell's equations in the field of electromagnetism.
Its application to mechanics happened later.