Let $\iota\colon X\subset \mathbb{P}^N$ be a quasi-projective variety. Denote by $\mathcal{O}:=\mathcal{O}_{\mathbb{P^n}}$ the sheaf of regular functions. I guess that the sheaf of regular functions $\mathcal{O}_X$ is isomorphic to the pullback sheaf $\iota^{-1}\mathcal{O}$, however I'm not able to find a morphism of sheaves $\mathcal{O}_X\longrightarrow \iota_{-1}\mathcal{O}$.
Can you help me?
It depends which pullback. If $i : X \to Y$ is a morphism, say the inclusion of a variety, $\mathcal O := \mathcal O_Y$ and $F$ is a sheaf on $Y$, we have two pullbacks : the sheaf-theoretic pullback $i^{-1}F$ and the algebraic-geometry pullback : $i^* F := i^{-1} F \otimes_{\mathcal i^{-1}\mathcal O} \mathcal O_X$, where $\mathcal O_X$ is a obvious $i^{-1} \mathcal O$-module (writing down the definition, you are essentially restricting a global function to $X$, in particular $\mathcal O_X$ is a $i^{-1}\mathcal O$ module).
When $F = \mathcal O$ you see indeed that $i^* \mathcal O = \mathcal O_X$ canonically.
If you take the other pullback this is false : for example when $X = x \in \Bbb P^n$ is a point, $i^{-1}\mathcal O = \mathcal O_{\Bbb P^n,x}$ (the stalk at $x$, which looks like power series at $x$) which is certainly not $\mathcal O_{x}$ (the structure sheaf of a point, which is a finite-dimensional vector space, of dimension $1$ if $x$ is a reduced point).