Thanks to Jan Bohr answer and comment I edited this question.

Let $E$ be a vector bundle , $E^*$ the dual bundle and $D$ a density bundle. Denote by $\Gamma(E)$ the space of section of the bundle $E$.

By definition the a distribution $\omega$ in a vector bundle $E$ is $\omega \in\Gamma(E^*\otimes D)^\prime$ where $\Gamma(E^*\otimes D)^\prime$ is the topological dual of $\Gamma(E^*\otimes D)$.

Now in this paper Algebraic Structure of Classical Field Theory: Kinematics and Linearized Dynamics for Real Scalar Fields by Romeo Brunetti, Klaus Fredenhagen, Pedro Lauridsen Ribeiro they define at page 14 the Gateaux derivative as $$F^K[\phi](\phi_1...,\phi_k)=\left<F^K[\phi],\phi_1\otimes...\otimes\phi_k\right>=\frac{d^k}{d\lambda_1...d\lambda_k }\left(\phi + \lambda_1\phi_1+...\lambda_k\phi_k\right)$$ where $<>$ denotes dual pairing and $\phi,\phi_1...,\phi_k \in\Gamma(E) $

Now they say that $F^K[\phi]$ is a distribuition . But $F^K[\phi]$ is not in $\Gamma(E^*\otimes D)^\prime$ how can it be a distribution?

Thanks to Jan Bohr answer and comment I add this part . In this paper Properties of field functionals and characterization of local functionals by Christian Brouder, Nguyen Viet Dang, Camille Laurent-Gengoux, Kasia Rejzner at page 10

they define $F^K[\phi]$ as a d distributional section on $M^k$ for a general vector bundle.

Maybe they are using $<>$ to identify $E$ with $E^*$ but I am not sure