We study \(\frac12\)-derivations of Lie algebras with values in modules, with emphasis on weight modules. For \(\mathfrak{sl}_2\), we prove a Casimir criterion for simple weight modules admitting nonzero \(\frac12\)-derivations. We use this criterion to determine the \(\frac12\)-derivations into highest weight modules, lowest weight modules, and dense weight modules. For finite-dimensional complex simple Lie algebras of rank at least two, we prove the vanishing of \(\frac12\)-derivations into Verma modules, cuspidal weight modules, and all infinite-dimensional simple weight modules with finite-dimensional weight spaces. This extends the \(\delta=\frac12\) case of the \(\delta\)-first Whitehead lemma of Zohrabi--Zusmanovich. We also determine all \(\frac12\)-derivations from the Witt algebra with values in tensor density modules.