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.