How do you solve the system $Ax=b$ where $\|x\|_1 \leq \delta$ and $x \in \lbrace 0,1 \rbrace^n$?

You are correct that this can be framed as a binary integer program. I do not recognize it as a problem that has its own name or "niche" within the more general area of integer linear programming. It can certainly be solved using any integer programming code, subject to the usual issue that a large enough dimension (referring to both $n$ and the dimension of $b$) may result in the solver running out of time or memory.

Note that the norm constraint $\vert\vert x \vert\vert_1 \le \delta$ can be rewritten as $\sum_{i=1}^n x_i \le \delta$.

There being no specified objective function, you could just minimize 0, or perhaps minimize (or maximize) $\sum_{i=1}^n x_i$.