If you make a gauge theory about $\mathrm{SU}(N)$, you also have one about $\mathrm{SO}(N)$ since $\mathrm{SO}(N) \subset \mathrm{SU}(N)$.
The representation theory of $\mathrm{SO}(N)$ is more complicated since it has not that many nice properties compared to $\mathrm{SU}(N)$ - the latter preserves orthogonal, complex and symplectic structures while the former only respects orthogonal structures. If you think this should imply that $\mathrm{SO}(N)$ is the larger group, this is because the symplectic and orthogonal structures that $\mathrm{SU}(N)$ preserves are in other dimensions, more precisely $$\mathrm{U}(N) = \mathrm{O}(2N) \cap \mathrm{GL}_\mathbb{C}(N) \cap \mathrm{Sp}_\mathbb{R}(2N)$$
Heuristically, working with less symmetry (i.e $\mathrm{SO}$ vs $\mathrm{SU}$) means more work, thus typical presentations prefer to work with the unitary groups. Also, most quantum theories are inherenty working with complex numbers, so a unitary (gauge) symmetry is more natural than a orthogonal real one.