I'm reasonably convinced the algebra generates all of $B(\mathcal{F}(H))$. Note that $1-\sum_{i=1}^\infty s_is_i^*$ converges strongly to the projection $P_0$ onto $\mathbb{C}$ and similarly the sum $1-\sum_{|\alpha|=n}s_\alphas_\alpha^*$ over multi-indices converges to the projection $P_n$ onto vectors with highest tensor power $n-1$.
Pick vectors $u\in H^{\otimes n}$, $v\in H^{\otimes m}$, then the operators $s_u$, $s_v$ (the obvious generalisations of the $s_i$) are in your algebra and so is the rank one operator $s_vs_u^*P_{n+1}=|v><\ u|$. Finally note you may approximate any rank one in $B(\mathcal{F}(H))$ by these.