A precise reference for this result is the book "The Subgroup Structure of the Finite Classical Groups" by P. Kleidman and M. Liebeck, Proposition 4.1.7.

The stabilizer of a non-singular $1$-space in the simple group $\Omega^{\pm}_{2n}(q)$ is isomorphic to ${\rm Sp}_{2n-2}(q)$. There is a unique conjugacy class of subgroups of this type, so it is normalized by the diagonal outer automorphism of $\Omega^{\pm}_{2n}(q)$, and the stabilizer in ${\rm SO}_{2n}(q) = {\rm GO}_{2n}(q)$ is isomorphic to ${\rm Sp}_{2n-2}(q) \times C_2$.

You asked about the stabilizer of a vector rather than a $1$-space, but that will be the same, because there are no scalars of order $2$ when $q$ is even.

A precise reference for this result is the book "The Subgroup Structure of the Finite Classical Groups" by P. Kleidman and M. Liebeck, Proposition 4.1.7.

The notation used for the various orthogonal groups varies a lot from book to book, so I am not completely sure what you mean by $O(n,q)$, but presumably $O'(n,q)$ is intended to denote the simple subgroup, which is often denoted $\Omega(n,q)$ or $\Omega_n(q)$.

The stabilizer of a non-singular $1$-space in the simple group $\Omega^{\pm}_{2n}(q)$ is isomorphic to ${\rm Sp}_{2n-2}(q)$. There is a unique conjugacy class of subgroups of this type, so it is normalized by the diagonal outer automorphism of $\Omega^{\pm}_{2n}(q)$, and the stabilizer in ${\rm SO}^{\pm}_{2n}(q) = {\rm GO}^{\pm}_{2n}(q)$ is isomorphic to ${\rm Sp}_{2n-2}(q) \times C_2$.

You asked about the stabilizer of a vector rather than a $1$-space, but that will be the same, because there are no scalars of order $2$ when $q$ is even.