Let $N$ be any closed manifold with nontrivial Stiefel-Whitney classes. This includes any non-orientable manifold or any manifold with odd Euler characteristic.
By Whitney's Embedding Theorem, $N$ embeds into $\mathbb{R}^n$ for $n$ large enough.
I claim that the orthonormal frame bundle of $N$ is not trivial. The point is that it on the vector bundle level, we have $T\mathbb{R}^n|_N = TN\oplus \nu $ where $\nu$ is the normal vector bundle. Since $\mathbb{R}^n$ has trivial tangent bundle, we have $w_k(\mathbb{R}^n) = 0$ for all $k$. Hence, $$0 = w_k(T\mathbb{R}^n|_N) = \sum_{i+j = k} w_i(TN)\cup w_j(\nu)\in H^k(N,\mathbb{Z}/2\mathbb{Z}).$$
Since $w(TN)$ is nontrivial by assumption, this impiles $w(\nu)$ is nontrivial as well, which implies $\nu$ is nontrivial as a bundle. Now one just notes that $\nu$ is the associated bundle to the principal bundle of orthonormal frames in $M$ orthogonal to $N$. Hence, the fact that $\nu$ is nontrivial implies that this principal bundle is non-trivial as well.