Let's look on the linearization of the action (it exists for a lie group $G$ acting on affine variety $X$). That way, $X$ is embedded into an affine space $V = \mathbb{A}^N$ as a closed subset, and $G$ acts linearly on $V$. Consider now an element $s \in \mathfrak{g}$ and the corresponding linear vector field on $V$, let $x \in X$ be its fixed point. $T_X(x)$ is an invariant subspace of $s$. Provided it would have an invariant complement, projection along this complement would give us the desired form.

For reductive $K$ we can do even better - $Stab_x \cong K$, and $K$ is reductive, so we can chose the invariant complement with respect to whole $K$. Projection along this complement on $T_x(X)$ gives the desired etale linearization for the whole stabilizer simultaneously.

Let's look on the linearization of the action (it exists for a lie group $G$ acting on affine variety $X$). That way, $X$ is embedded into an affine space $V = \mathbb{A}^N$ as a closed subset, and $G$ acts linearly on $V$. Consider now an element $s \in \mathfrak{g}$ and the corresponding linear vector field on $V$, let $x \in X$ be its fixed point. $T_X(x)$ is an invariant subspace of $s$. Provided it would have an invariant complement, projection along this complement would give us the desired form.

For reductive $K$ we can do even better - $Stab_x \cong K$, and $K$ is reductive, so we can chose the invariant complement with respect to whole $K$. Projection along this complement on $T_x(X)$ gives the desired etale linearization for the whole stabilizer simultaneously.

EDIT: actually I've realised that you ask about somewhat strange normal form. I read it automatically as linear vector field ($\Sigma a_{ij}x^i \partial_j$). The rest is linear algebra anyways (the form you are demanding as it is written I think does not exist even for linear vector fields).

Let's look on the linearization of the action (it exists for reductive group $G$ acting on affine variety $X$). That way, $X$ is embedded into an affine space $V = \mathbb{A}^N$ as a closed subset, and $G$ acts linearly on $V$. Consider now an element $s \in \mathfrak{g}$ and the corresponding linear vector field on $V$, let $x \in X$ be its fixed point. $T_X(x)$ is an invariant subspace of $s$. Provided it would have an invariant complement, projection along this complement would give us the desired form.

For reductive $K$ we can do even better - $Stab_x \cong K$, and $K$ is reductive, so we can chose the invariant complement with respect to whole $K$. Projection along this complement on $T_x(X)$ gives the desired etale linearization for the whole stabilizer simultaneously.

Let's look on the linearization of the action (it exists for a lie group $G$ acting on affine variety $X$). That way, $X$ is embedded into an affine space $V = \mathbb{A}^N$ as a closed subset, and $G$ acts linearly on $V$. Consider now an element $s \in \mathfrak{g}$ and the corresponding linear vector field on $V$, let $x \in X$ be its fixed point. $T_X(x)$ is an invariant subspace of $s$. Provided it would have an invariant complement, projection along this complement would give us the desired form.

For reductive $K$ we can do even better - $Stab_x \cong K$, and $K$ is reductive, so we can chose the invariant complement with respect to whole $K$. Projection along this complement on $T_x(X)$ gives the desired etale linearization for the whole stabilizer simultaneously.