Let $X:=(X_1,\dots,X_n)$, where the $X_j$'s are iid copies of $\zeta$. Then the problem is about conditions for $$P(\|X\|\le u\sqrt n)\le C^n u^n\tag{1}$$ for some real $C>0$, all natural $n$, and all small enough real $u>0$ (and then for all real $u>0$, possibly with a different $C>0$).
If the distribution of $X_1$ has a density bounded by some real $c>0$, then the distribution of $X$ has a density bounded by $c^n$, so that $$P(\|X\|\le u\sqrt n)\le c^n (u\sqrt n)^n|B_n|\le C^n u^n$$ for some real $C>0$, all real $u>0$, and all natural $n$, where $|B_n|$ is the volume of the unit ball $B_n$ in $\mathbb R^n$; so, (1) holds.
On the other hand, (1) cannot hold (for $C,u,n$ specified above) if the distribution of $X_1$ has an atom at $0$. Moreover, (1) cannot hold for any real $C>0$, any natural $n$, and all small enough real $u>0$ if the distribution of $X_1$ has a density $p$ such that $p(x)\to\infty$ as $x\to0$. Indeed, then for each natural $n$ we have $$P(\|X\|\le u\sqrt n)/u^n=\int_{u\sqrt n\,B_n}\prod_{j=1}^n (p(x_j)\,dx_j)/u^n\to\infty$$ as $u\downarrow0$.
