Relativized world where $L^A=NP^A$
My question is just whether there is a known relativization barrier.


Yes, there is a known relativization barrier. It's given by $A:=TQBF$, because Emil Jeřábek (see comments) is right: the statement that $TQBF$ is $PSPACE$-complete under logspace many-one reductions is well known.



Emil Jeřábek's remark "(or even uniform $\mathbf{AC}^0$)" seems less relevant, because $\mathbf{AC}^0$ is easily separated from $L$ (or even uniform $\mathbf{NC}^1$). Because the resulting formula actually depends only very locally on the input for the $PSPACE$-Turing machine, one could even claim that $TQBF$ is $PSPACE$-complete under $L$-uniform $\mathbf{NC}^0$ (instead of uniform $\mathbf{AC}^0$). But I find this misleading, because the circuit doesn't really do any non-trivial work at all, it basically just writes down a template.

(The reduction writes down $n^k$ times $\exists c_{1.5}\forall (c_3, c_4) \in \{(c_1, c_{1.5}), (c_{1.5}, c_2)\}$ with systematically varying indices, where each $c_i$ are actually $n^k$ Boolean variables. This entire part only depends on the length of the input, hence the corresponding output variables are hardwired to constants. The final formula could be written as $\forall z. (z_1=i_i\land\dots\land z_n=i_n)\land \phi(x,y,z)$ such that only $i_i,\dots,i_n$ depends on the actual input, and such that this is the only place where $i_i,\dots,i_n$ appear. So this is nearly trivially an $\mathbf{NC}^0$, the entire work has been done by the $L$-algorithm writing down the circuit, whose outputs are nearly all directly hardwired to constant.)