The situation for graded modules over a pair of Koszul dual algebras is more complicated, actually. What the question says is true for Koszul algebras $A$ and $A^!$ provided that $A$ is Noetherian and $A^!$ is finite-dimensional (including the case of the symmetric and exterior algebras) but not otherwise. In general one can say that the unbounded derived categories of positively graded modules with finite-dimensional components over $A$ and $A^!$ are anti-equivalent. The subcategories of complexes of positively graded modules bounded separately in every grading in these unbounded derived categories are also anti-equivalent.
One can replace the contravariant anti-equivalence with a covariant equivalence by considering positively graded modules over one of the algebras and negatively graded modules over the other one (both algebras being considered as positively graded). In this case one does not have to require the components of the modules to be finite-dimensional.
With algebras over operads, the analogue of the equivalence for graded modules involves DG-algebras with an additional positive grading (there being only the ground field $k$ in the additional grading $0$ and nothing in the negative additional grading), with the additional grading preserved by the differential. The Koszul duality is an anti-equivalence between the localizations of the categories of DG-algebras of this kind, with every component of fixed additional grading being a bounded complex of finite-dimensional vector spaces, by quasi-isomorphisms. For some operads (e.g., for Lie and Com) one has to assume the field $k$ to have characteristic $0$, while for some others (e.g., Ass) one doesn't.
If one wishes to replace the contravariant anti-equivalence with a covariant equivalence in the case of algebras over operads, one has to consider algebras on one side of the equivalence and coalgebras on the other side. Then the boundedness and finite-dimensionality requirements can be dropped.
What I've described above is the homogeneous Koszul duality; the nonhomogeneous case (with ungraded modules or algebras without the additional grading) is more complicated, though also possible. See my answer to the question linked to from the question above.
References: 1. Beilinson, Ginzburg, Soergel "Koszul duality patterns in representation theory", 2. My preprint "Two kinds of derived categories, ...", arXiv:0905.2621, Appendix A.
One standard answer*, in which any reasonable (characteristic $0$ — I haven't thought about any other case) algebraic category can be given a simplicial structure, is the following.
Let $\mathbb Q[\Delta^k] = \mathbb Q[t_0,\dots,t_k,\partial t_0,\dots,\partial t_k] / \bigl\langle \sum t_i = 1,\ \sum\partial t_i = 0\bigr\rangle$ denote the differential graded commutative algebra (dgca) of polynomial forms on the standard $k$-simplex. Here $t_i$ are in (co)homological degree $0$, and their derivatives $\partial t_i$ are in degree $\pm 1$ depending on whether you prefer homological or cohomological conventions. It is straightforward to check that $\mathbb Q[\Delta^k]$ has (co)homology only in degree $0$, where it is $1$-dimensional. Moreover, there are natural face and degeneracy maps between different $\mathbb Q[\Delta^k]$, making $\mathbb Q[\Delta^\bullet]$ into a simplicial dgca.
Given two $L_\infty$ algebras $V,W$ (or, really, objects of any reasonable category of "algebras"), one then defines the space of maps $V \to W$ to be the simplicial set
$$ \hom_\bullet(V,W) = \hom(V,W[\Delta^\bullet]),$$
where $W[\Delta^\bullet] = W\otimes_{\mathbb Q} \mathbb Q[\Delta^\bullet]$ is the $L_\infty$ algebra $W$ base-changed to live over the $k$-simplex. It is reasonably straightforward to prove that this simplicial set satisfies the Kan horn-filling condition, at least when $V$ is "quasifree" — in particular, in your situation of "nonlinear $L_\infty$-algebra homomorphisms", the Kan condition is always satisfied.
Before I spell this out, I'm going to change your notation. What you called $f_k$ I will call $f^{(k)}$, since it plays the role of the "$k$th Taylor coefficient of $f$". That way, I can ask "what is a homotopy between two morphisms $f_0,f_1 : V \to W$ of $L_\infty$-algebras?"
The answer is the following data: (1) a (nonlinear) homomorphism $f_t: V \to W$ that depends polynomially on a parameter $t$, with the correct evaluations $f_t|_{t=0} = f_0$ and $f_t|_{t=1} = f_1$; (2) maps $\phi^{(k)}_t : V \to W[1]$ (or maybe I mean $[-1]$), also depending polynomially on the parameter $t$. These data must satisfy a certain ODE of the form:
$$ \frac{\mathrm d}{\mathrm d t} f_t = \operatorname{ad}_{f_t}(\phi_t) $$
Of course, this is really an infinite sequence of equations (which are equations to things that depend polynomially on $t$). The $k$th entry on the left hand side is $ \frac{\mathrm d}{\mathrm d t} f_t^{(k)}$. On the right hand side, the $k$th entry is computed as follows (up to a sign which I don't feel like working out). Consider the equations saying that $f_t$ is a homomorphism; one of these equations is an equation of things with $k$ inputs $x$. Sum over all ways to replace, in each summand in this equation, one of the occurrences of an $f$ by a $\phi$. Such a sum is what I mean by the right-hand side. In short-hand, what I mean is: there is (a sequence of) equations $M(f)$, such that $f$ is a homomorphism iff $M(f) = 0$. The right hand side is $\frac{\partial M}{\partial f} \cdot \phi$.
In good situations like yours, all the ODEs that occur when studying $\hom_\bullet(V,W)$ are pretty well behaved. In particular, their integral forms are contraction mappings in the appropriate sense, so the initial and boundary value problems are pretty easy to analyze formally.
*Here is an important (elementary) exercise to work out if you want to understand this "standard answer." Consider just the category of chain complexes. Then, for $k \geq 0$, $\pi_k\bigl( \hom(V,W[\Delta^\bullet])$ is the space of chain maps $V \to W[\pm k]$ modulo chain homotopies, i.e. it is $\mathrm{H}_k(\underline\hom(V,W))$, where $\underline\hom$ denotes the chain complex of all linear maps $f: V \to W$ with differential $f \mapsto [\partial,f] = \partial_W\circ f -(-1)^{\deg f}f\circ \partial_V$. (Whether the shift should be $[k]$ or $[-k]$, and whether I mean $\mathrm H_{\pm k}$ or $\mathrm H^{\pm k}$ or ..., depend on your conventions, so I didn't work them out.)
Best Answer
It is a typo. The map $f$ should only be assumed to be a morphism of the underlying graded $\mathbb S$-modules.