Let $L$ be some first-order language and let $T$ be a collection of formulas. Assume then that for all structures $M$, we have a $\phi\in T$ s.t. $M\models \phi$. I'm trying to show that this implies that there's a finite collection $\phi_1,\ldots,\phi_n\in T$ s.t. $\models \phi_1\vee \cdots \vee \phi_n$.
The only idea I've had so far is showing that $T$ must be inconsistent, so that it must have a finite inconsistent subset. However, I don't see how.
You are right, the result is a consequence of the Compactness Theorem.
Suppose that there is no set $\{\phi_1, \dots,\phi_n\}$ with the property described in the question. This means that given any $\phi_1,\phi_2,\dots,\phi_n$ in $T$, the set $\{\lnot\phi_1, \lnot\phi_2,\dots,\lnot \phi_n\}$ is consistent.
Consider the collection $T^\ast$ of all $\lnot \phi$, where $\phi$ ranges over $T$. Every finite subset of $T^\ast$ has a model, so $T^\ast$ does. Let $M$ be a model of $T^\ast$. Then there is no $\phi\in T$ such that $\phi$ is true in $M$.