From work of Clouatre and Ramsey, we now have a robust notion of residual finite dimensionality for possibly non-self-adjoint operator algebras. In works of Clouatre and the speaker, as well as upcoming work of Hartz, we also have a good understanding of RFD in terms of representations of the operator algebra. The question of whether certain concrete operator algebras are RFD is known to be highly difficult, but for reduced group C*-algebras a result Bekka and Louvett states that they are RFD if and only if the group is amenable and RF. Following previous work of the speaker with Clouatre, we prove residual finite dimensionality for large classes of left-regular operator algebras of semigroups. Our proofs use machinery from recent work of Laca and Sehnem, on the existence of a universal C*-algebras associated to semigroups that embed into groups.