I can tell you that Lagrange does not use any name for the symmetric functions in his big paper
Re/flexion sur la Re/solution alge/brique des e/quations. Nouveau Me/moires de l'Acade/mie royale des Sciences et Belles-Lettres de Berlin, anne/es 1770 et 1771. Works vol. 3, 205-421.
Vandermonde uses the name "type de combinaison" mostly shortened to "type" for homogeneous symmetric polynomials. He also uses the name "type partiel" for the sum of monomials in n indeterminates, where the summands are the images of one of the monomials under the powers of a permutation. (Gauss' sums of his periods!)
Me/moire sur la re/solution des e/quations. Histoire de l'Acade/mie royale des Sciences. Anne/e 1771. Paris 1774, 365-416. The paper was read before the Academy in 1770.
Vandermonde was not yet "de l'Acade/mie" in 1770. That's why the paper was not published in the Histoire of 1770.
So, you have to look for publications in the period from 1770 to 1799.
Concerning Vandermonde and Gausz, see
Henri Lebesgue, L'OEuvres mathe/matiques de Vandermonde. L'Enseignement de mathe/matique I, 203--223, 1955