What I had in mind was simply that some proof systems lend themselves to systematic proof search which, if $\models A$, is guaranteed to be successful. E.g., in sequent calculus LK, you basically apply the rules in reverse, and if you do it systematically (in particular, apply the strong quantifier rules often enough), you eventually find a derivation. This is the basis of the direct completeness proof for LK, I believe first due to Schütte. Completeness is proved this way in Takeuti's proof theory book; I'm sure there are more readily available references but none come to mind off the top of my head. As @Mark Savig mentioned, any such derivation in the sequent calculus LK can be translated into one in natural deduction or Hilbert-style systems. You can do systematic proof search in natural deduction directly as well, but Fitch-style is not the best notation to do that in. But essentially the "proof construction strategies" in intro textbooks that use Fitch-style proofs are such proof search methods.
(Asides: 1. Putting this as "it is possible to construct a derivation" is perhaps misleading as this question shows; I'll clarify this in the text. 2. The Henkin-style completeness proof essentially also includes such a proof search procedure, but the proof obscures this. Basically, if your set is inconsistent, the Lindenbaum construction will fail to produce a maximally consistent set, and the derivation of an explicit contradiction from some stage $\Gamma_i$ will give you a derivation of a contradiction from the starting set $\Gamma$. But that is very roundabout. 3. Proof search is possible for every proof systems where the derivations are computably enumerable, and is guaranteed to terminate if the system is complete. But that it is complete can't be shown this way except if you can construct a countermodel from an infinite failed proof search, like in Schütte's proof. And this requires some non-constructive principle such as WKL.)