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 real $u>0$, and all natural $n$. 

Clearly, (1) cannot hold (for such $C,u,n$) if the distribution of $X_1$ has an atom at $0$. 

On the other hand, 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 ][1] in $\mathbb R^n$. So, (1) holds. 


  [1]: https://en.wikipedia.org/wiki/Volume_of_an_n-ball#The_volume