Let $(k_n)_{n \in \mathbb{N}}$ be a sequence of positive integers growing to infinity at a sublinear rate, $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$. Given a sequence of $n$-dimensional random vectors $\{Y^{(n)}\}_{n \in \mathbb{N}}$ belonging to a certain class, which includes uniform distributions on suitably scaled $\ell_p^n$-balls or $\ell_p^n$-spheres, $p \geq 2$, and product distributions with sub-Gaussian marginals, we study the large deviations behavior of the corresponding sequence of $k_n$-dimensional orthogonal projections $n^{-1/2} \boldsymbol{a}_{n,k_n} Y^{(n)}$, where $\boldsymbol{a}_{n,k_n}$ is an $(n \times k_n)$-dimensional projection matrix lying in the Stiefel manifold of orthonormal $k_n$-frames in $\mathbb{R}^n$. For almost every sequence of projection matrices, we establish a large deviation principle (LDP) for the corresponding sequence of projections, with a fairly explicit rate function that does not depend on the sequence of projection matrices. As corollaries, we also obtain quenched LDPs for sequences of $\ell_2$-norms and $\ell_\infty$-norms of the coordinates of the projections. Past work on LDPs for projections with growing dimension has mainly focused on the annealed setting, where one also averages over the random projection matrix, chosen from the Haar measure, in which case the coordinates of the projection are exchangeable. The quenched setting lacks such symmetry properties, and gives rise to significant new challenges in the setting of growing projection dimension. Along the way, we establish new Gaussian approximation results on the Stiefel manifold that may be of independent interest. Such LDPs are of relevance in asymptotic convex geometry, statistical physics and high-dimensional statistics.