There is an exponential upper bound of $9^n$, since every vertex of $B_1 \cap B_2$ is the intersection of a $k$-face of $B_1$ and a $(n-k)$-face of $B_2$ for some $k$, and the $\ell_1$-ball has $3^n$ faces (all dimensions counted together).
If you allow rotated balls, you cannot do better for large $n$ except for the value of the constant $9$. Indeed the intersection $K$ of the $\ell_1$ ball with a randomly rotated copy of itself has (with high probability) the property that $\frac{1}{\sqrt{n}} B \subset K \subset \frac{C}{\sqrt{n}} B$ for some constant $C$ (here $B$ is the Euclidean unit ball). This is known as the "global form of Kashin's theorem", see Theorem 5.5.4. in [1]. That sandwiching forces $K$ to have at least $\exp(cn)$ vertices for some other constant $c$ (essentially because a spherical cap of fixed angle less than $\pi/2$ covers an exponentially small proportion of the sphere as $n \to \infty$).
[1] S. Artstein-Avidan, A. Giannopoulos, V. Milman, Asymptotic Geometric Analysis, Part I