Can the Bernstein-Vazirani algorithm be applied to a remote classical function? As in, a function we are not able to set up an oracle for and is completely classical?
The answer is no. The Bernstein-Vazirani algorithm requires that the use of a quantum oracle that is structured based on the problem’s parameters - namely, the string $s$.
The classical strategy we can use in the classical oracle case is to simply call the function $n$ times (where $n$ in the number of bits needed to represent $s$), each time with a different bit set to $1$, while all other bits are set to $0$. This is different than in the quantum case, where we only need one query to the oracle.