Let \((R,\mfm)\) be a Noetherian local ring and let \(M\) be a finitely generated flat \(R\)-module. Then \(M\) is free.
Let \(k=R/\mfm\) and \(n=\dim_kM/\mfm M\). By Nakayama's lemma for a local ring, lifts of a \(k\)-basis of \(M/\mfm M\) give a surjection \(R^n\to M\); let \(K\) be its kernel, which is finitely generated because \(R\) is Noetherian. Since \(M\) is flat, \(\Tor_1^R(k,M)=0\), so applying \(k\otimes_R-\) to \(0\to K\to R^n\to M\to0\) gives an exact sequence \[ 0\to K/\mfm K\to k^n\to M/\mfm M\to0. \] The map \(k^n\to M/\mfm M\) is a surjection between \(k\)-vector spaces of dimension \(n\), hence an isomorphism, so \(K/\mfm K=0\). By Nakayama's lemma, \(K=0\), and \(M\cong R^n\).
Applied to stalks, this shows that a coherent sheaf \(\mathcal F\) on a Noetherian scheme \(X\) whose stalks \(\mathcal F_x\) are flat \(\OO_{X,x}\)-modules has free stalks, and is therefore locally free, that is, the sheaf of sections of a vector bundle.