In the lecture notes I'm currently reading, the professor wrote that an application of the Löwenheim-Skolem theorem (stated below) is to show that the theory of infinite vector spaces over a field $K$ in the language $\mathcal{L} = (0, +, (f_q)_{q \in K})$ is complete. Here $f_q$ are the unary functions to be understood like scalar-multiplication by $q$ in the vector space.
My confusion is: The Löwenheim-Skolem theorem makes a statement about the existence of models of a certain size. Why would that help? To apply Löwenheim-Skolem I need the theory to be consistent anyway - and can't I then just directly apply Gödel's completeness theorem?
Theorem (Löwenheim-Skolem (ZFC)): Let $T$ be a consistent theory with infinite models in a language $\mathcal{L}$ of cardinality $\kappa$. Then for any infinite cardinal $\lambda \geq \kappa$ the theory $T$ admits a model of size $\lambda$.
First let me address your confusion over why Lowenheim-Skolem is necessary. The theory of a class $\mathbb{K}$ of structures is the set of sentences satisfied by all structures in that class. Unlike the theory of a single structure, this theory need not be complete - for instance, if $\mathbb{K}$ is the class of groups, then the commutativity axiom "$\forall x, y(x*y=y*x)$" is not in $Th(\mathbb{K})$ (since there are non-abelian groups), but neither is its negation (since there are abelian groups).
So we can't just automatically conclude that the theory in your question is complete; we need to argue that any two structures in your class are "basically the same" (= elementarily equivalent). A standard trick for doing this is: given $\mathcal{A}$ and $\mathcal{B}$ which we want to show are elementarily equivalent, build "bigger" structures $\mathcal{A}'$ and $\mathcal{B}'$, such that we know $\mathcal{A}\equiv\mathcal{A}'$ and $\mathcal{B}\equiv\mathcal{B}'$, and show that $\mathcal{A}'\cong\mathcal{B}'$. Do you see a way to use Lowenheim-Skolem to do this, here? HINT: what determines the isomorphism type of a vector space? How is this related to its cardinality?
Incidentally, re: the strategy above, a neat theorem of Keisler and Shelah gives a way to do this that always works: if $\mathcal{A}\equiv\mathcal{B}$, then they have isomorphic ultrapowers (and obviously conversely)! This is Theorem 10.7 of this survey article on ultraproducts by Keisler, which is valuable reading if you're interested in the more set-theoretic side of model theory.