(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 of $\widetilde Z(\widetilde G_{\mathbb C}=\mu_2$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$4r$. 1 (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 of$\widetilde G_{\mathbb C}=\mu_2$, 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\$.