The whole flag variety is definitely not a single $O(n)$-orbit. You can see this already in the case of $n=2$, where the flag variety is just $\mathbb P^1$, and there are two orbits: $\{\pm i\}$ and everything else. More generally, by Witt's theorem, the $O(n)$-orbit on the grassmannian of $r$-planes through any given totally isotropic subspace of dimension $r$ is the (closed!) subvariety of all totally isotropic subspaces of dimension $r$. So you will have 'the' variety of isotropic flags as a closed $O(n)$-orbit on the flag variety. What's going on here is that a closed $O(n)$-orbit in the flag variety will be projective, hence will look like $O(n)/P$ for some parabolic, i.e. it will be a partial flag variety for $O(n)$. The same kind of thing will be true for $Sp(n)$, with Lagrangian subspaces taking the place of isotropic subspaces.

Anyway, there is a unified framework for dealing with these types of questions. The key observation is that $O(n)$ and $Sp(n)$ (if $n$ is even) are *symmetric subgroups* of $GL(n)$, i.e., each can be realized as the set of fixed points of an algebraic involution $\theta \colon GL(n) \to GL(n)$, namely $\theta(g)=g^{-t}$ and $\theta(g)=Jg^{-t}J^{-1}$, resp., where $J$ is the matrix of the symplectic form defining $Sp(n)$. Your question is thus a special case of the following:

If $K=G^\theta$ is a symmetric subgroup of $G$, what can we say about the $K$-orbits on $G/B$?

The answer is: lots! A good starting point is the set of notes to Allen Knutson's course on the topic. Be sure to also take a look at the Richardson--Springer paper linked to on that page.