(I add details to my comments.) The answer depends on $n=4r$. Write $G=Sp(r)/\mu_2$. If $r=1$, then $G\simeq SO_3$, so $G$ admits a faithful 4-dimensional representation into $SO_4$. Similarly, if $r=2$, then $G\simeq SO_5$, hence $G$ admits a faithful 8-dimensional representation into $SO_8$. (Of course, in these cases $S^{4r-1}$ is not an orbit.) For $r\ge 3$ the group $G$ has no nontrivial representations of dimension $4r$, see below, hence it cannot be embedded into $SO_{4r}$.

An irreducible real $n$-dimensional representation of the real algebraic group $G$ induces an irreducible complex $n$-dimensional representation
of $G_{\mathbb C}=Sp_{r,{\mathbb C}}/\mu_2$.
The irreducible complex representations of the simple group
$\widetilde G_{\mathbb C} =Sp_{r,{\mathbb C}}$
of type $C_r$
for $r>1$ of dimension $n<{\rm dim}\ \widetilde{G}_{\mathbb C}$
are listed in the paper of Andreev, Vinberg, and Elashvili,
Table 1.
They are the fundamental irreducible representations $R(\pi_1)$ of dimension $2r$, $R(\pi_2)$ of dimension $2r^2-r-1$,
and, for $r=3$, $R(\pi_3)$ of dimension 14. Since the representations $R(\pi_1)$ and $R(\pi_3)$
are nontrivial on the center $Z(\widetilde G_{\mathbb C})\simeq\mu_2$,
we see that the only nontrivial irreducible representation of $G_{\mathbb C}$ of dimension $n<{\rm dim}\ G_{\mathbb C}$
is the representation $R(\pi_2)$ of dimension $2r^2-r-1$,
hence $R(\pi_2)$ is the irreducible representation of $G_{\mathbb C}$ of the smallest dimension.
For $r\ge 3$ we have $2r^2-r-1>4r$, hence $G_{\mathbb C}$ has no nontrivial representations of dimension $4r$.