Consider $\Sigma$ a compact surface embedded into a compact 3-manifold, such that $\Sigma$ is diffeomorphic to $\mathbb{R}\mathbb{P}^2$ (real projective plane) and $\varphi:\mathbb{S}^2 \to \Sigma$ is a local diffeomorphism such that $\varphi(p) = \varphi(-p)$, for every $p \in \mathbb{S}^2$.
How can I show that the pullback of the normal bundle N$\Sigma$ by the application $\varphi$ is a trivial bundle?
There are precisely two real line bundles over $\Bbb{RP}^2$: the trivial bundle and the canonical line bundle. So you just need to check that these pull back to the trivial bundle. But the trivial bundle obviously does; and you can prove that the canonical line bundle pulls back to the trivial bundle by considering $\varphi^* \kappa$ is the normal bundle of $S^2$ in $\Bbb R^3$. (Do this from the definition of the canonical bundle.)
E: Or just note that there are no nontrivial line bundles on $S^2$? Either because you know that real line bundles are classified by $H^1(X;\Bbb Z/2)$, or because you know that real line bundles on $S^2$ are classified by the homotopy class of their clutching function $S^1 \to GL_1(\Bbb R) = \Bbb R^\times$, and this has contractible components.