While local projections (LPs) are widely used for impulse response analysis, existing Bayesian approaches face fundamental challenges because a set of LPs does not constitute a likelihood function. Prior studies address this issue by constructing a pseudo-likelihood, either by treating LPs as a system of seemingly unrelated regressions with a multivariate normal error structure or by applying a quasi-Bayesian approach with a sandwich estimator. However, these methods lead to posterior distributions that are not "well calibrated," preventing proper Bayesian belief updates and complicating the interpretation of posterior distributions. To resolve these issues, we propose a novel quasi-Bayesian approach for inferring LPs using the Laplace-type estimator. Specifically, we construct a quasi-likelihood based on a generalized method of moments criterion, which avoids restrictive distributional assumptions and provides well-calibrated inferences. The proposed framework enables the estimation of simultaneous credible bands and naturally extends to LPs with an instrumental variable, offering the first Bayesian treatment of this method. Furthermore, we introduce two posterior simulators capable of handling the high-dimensional parameter space of LPs with the Laplace-type estimator. We demonstrate the effectiveness of our approach through extensive Monte Carlo simulations and an empirical application to U.S. monetary policy.