Primordial density perturbations in the radiation-dominated era of the early Universe are expected to generate stochastic gravitational waves (GWs) due to nonlinear mode coupling. In this emph{Letter}, we report on a search for such a stochastic GW background in the data of the two LIGO detectors during their second observing run (O2). We focus on the primordial perturbations in the range of comoving wavenumbers $10^{16}-10^{18}~{rm Mpc}^{-1}$ for which the stochastic background falls within the detectors sensitivity band. We do not find any conclusive evidence of this stochastic signal in the data, and thus place the very first GW-based constraints on the amplitude of the power spectrum at these scales. We assume a lognormal shape for the power spectrum and Gaussian statistics for the primordial perturbations, and vary the width of the power spectrum to cover both narrow and broad spectra. Derived upper limits ($95%$) on the amplitude of the power spectrum are $0.01-0.1$. As a byproduct, we are able to infer upper limits on the fraction of the Universes mass in ultralight primordial black holes ($M_mathrm{PBH} simeq 10^{-20}-10^{-19}M_{odot}$) at their formation time to be $lesssim 10^{-25}$.
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