The "Elements" of modern lattice theory, from which the terminology spread is Birkhoff's Lattice Theory (1940). After defining lattices and introducing cups and caps (rather than wedges and vees) for inf and sup he says the following on the subject of meets and joins (p.17):
"A large fraction of the most important partly ordered systems
considered in mathematics are lattices. Moreover in these systems, the opera
tions $\frown$ and $\smile$ usually correspond to familiar and significant constructions.
Example 1. Let $\Sigma$ consist of all the subsets of any aggregate $I$, and let inclusion mean set-inclusion. Then "joins" are sums of sets and "meets" are
Example 3. Let $\Sigma$ consist of the subgroups of any group, and let inclusion
mean set-inclusion. Then the terms "join" and "meet" have their usual meaning (also called union and intersection).
In other words, "join" joins (unites) subgroups together, and "meet" is where they meet (intersect). It is curious that Birkhoff considers the "usual meaning" to be for subgroups, not for sets. He does not credit anybody for the names, although he does mention that the definition is due to Peirce, and that Whitney suggested "cap" and "cup" names (Whitney himself used them for chains and cochains in homology).
The earliest appearance of meets and joins I found is in Birkhoff's first paper on lattices, On the combination of subalgebras (1933), where they are applied to subalgebras, subgroups and subrings, and again not credited to anybody. Birkhoff wrote this paper before he became aware of Dedekind's work. Von Neumann picked them up from Birkhoff in his later 1930-s papers. They do not appear in Peirce's On the Algebra of Logic (1885), or in Klein's expository paper Einige distributive Systeme in Mathematik und Logik (1929) that mentions many examples and names like conjuction/disjunction or union/intersection. Menger uses Summe and Durchschnitt (average) in Bemerkungen zu Grundlagenfragen (1928), but his 1936 New Foundations of Projective and Affine Geometry already has joins. I am going to speculate that Birkhoff was the terms' originator.
The beginnings of lattice theory go back to Boole, Dedekind and Peirce, but they did not have much of a follow-up until Emmy Noether revived Dedekind's work in her reshaping of algebra. Bilova in Lattice Theory – Its Birth and Life says that Birkhoff himself came up with the term "lattice" (under the influence of Hasse's drawings):
"Lattice structures started to be studied again at the end of 1920’s –
this time in still another area of mathematics. Karl Menger presented
the set of axioms characterizing projective geometries which are
in fact complemented modular lattices. His investigations did not attract
immediate attention, however, lattice structures soon appeared
also in the field of formal logic (Fritz Klein who gave lattices its German
name: “Verband”) and mainly algebra (Robert Remak, Oystein
Ore). The biggest merits in the early developments of lattice theory
belong to Garrett Birkhoff who also approached it from the side of
algebra and united its various applications. In his first article about
lattice structures  he rediscovered, apart from others, Dedekind’s
results, and only after its publishing it was revealed that the studies
of dual groups are identical with Birkhoff’s approach. G. Birkhoff
also introduced the English word “lattice”, which is not the translation
of its German equivalent, but was inspired by the image of some Hasse
diagrams presenting lattices."
The  is the aforementioned 1933 paper. I should also mention that aside from lattice theory meet and join are now commonly used in modern geometric algebra/exterior calculus (not to be confused with van der Waerden's term for Euclid's book II), where they are applied to subspaces and Grassmann's multivectors. This was Rota's contribution, see his with co-authors On the exterior calculus of invariant theory, and The Many Lives of Lattice Theory.