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For interacting fields, if one wants to keep close to what physicists do and steer clear of constructive methods a la Glimm-Jaffe (which is of course a better choice from the viewpoint of rigor but severely limits the models one may study in its present state of the art), one has to recourse to formal perturbation theory, which means one has to work with formal power series in the coupling constant and Planck's constant. This also means abandoning C$*$-algebras and working with more general *-algebras. Once one accepts this, perturbative renormalization can be dealt with in a rigorous way, using a language close to the one adopted above. See for instance R. Brunetti, M. Dütsch and K. Fredenhagen, "Perturbative algebraic quantum field theory and the renormalization groups", Adv. Theor. Math. Phys. 13 (2009) 1541–1599, arXiv:0901.2038 [math-ph].

For interacting fields, if one wants to keep close to what physicists do and steer clear of constructive methods a la Glimm-Jaffe (which are of course a better choice from the viewpoint of rigor but severely limit the models one may study in their present state of the art), one has to recourse to formal perturbation theory, which means one has to work with formal power series in the coupling constant and Planck's constant. This also means abandoning C$*$-algebras and working with more general *-algebras. Once one accepts this, perturbative renormalization can be dealt with in a rigorous way, using a language close to the one adopted above. See for instance R. Brunetti, M. Dütsch and K. Fredenhagen, "Perturbative algebraic quantum field theory and the renormalization groups", Adv. Theor. Math. Phys. 13 (2009) 1541–1599, arXiv:0901.2038 [math-ph].

• Vector space operations: $(\alpha W_f+\beta W_{f'})(\phi)=\alpha W_f(\phi)+\beta W_{f'}(\phi)$, $\alpha,\beta\in\mathbb{C}$;
• Involution: $(W^*_f)(\phi)=W_{-f}(\phi)$;
• Identity: $\mathbb{1}=W_0$;
• Weyl product: $(W_f W_{f'})(\phi)=e^{\frac{i}{2}\Delta_{m,g}(f\otimes f')}W_{f+f'}(\phi)$, where $\Delta_{m,g}$ is the distributional kernel of the difference between the retarded and the advanced fundamental solutions of the Klein-Gordon operator $\Box_g+m^2$. We recall that $\Delta_{m,g}$ provides our antisymmetric bilinear form, for $$\Delta_{m,g}(f\otimes f')=-\Delta_{m,g}(f'\otimes f)\ .$$

• Vector space operations: $(\alpha W_f+\beta W_{f'})(\phi)=\alpha W_f(\phi)+\beta W_{f'}(\phi)$, $\alpha,\beta\in\mathbb{C}$;
• Involution: $W^*_f=W_{-f}$;
• Identity: $\mathbb{1}=W_0$;
• Weyl product: $W_f W_{f'}=e^{\frac{i}{2}\Delta_{m,g}(f\otimes f')}W_{f+f'}$, where $\Delta_{m,g}$ is the distributional kernel of the difference between the retarded and the advanced fundamental solutions of the Klein-Gordon operator $\Box_g+m^2$. We recall that $\Delta_{m,g}$ provides our antisymmetric bilinear form, for $$\Delta_{m,g}(f\otimes f')=-\Delta_{m,g}(f'\otimes f)\ .$$

The Weyl algebra construction can be done abstractly for any real vector space (even infinite-dimensional) endowed with an antisymmetric bilinear form, thanks to B. Blackadar's universal C*-algebra construction using generators and relations ("Shape theory for C∗-algebras", Math. Scand. 56 (1985) 249–275). However, since you asked for a concrete correspondence, here is a more explicit description. In the case of (free) real scalar fields $\phi$ in a globally hyperbolic space-time $(M,g)$ subject to the Klein-Gordon equation $$ \Box_g\phi+m^2\phi=0\ ,$$ the Weyl unitaries are concretely given by the functionals $$ \mathscr{C}^\infty(M,\mathbb{R})\ni\phi\mapsto W_f(\phi)=e^{i\int_M f\phi\ d\mu_g}\ ,\quad f\in\mathscr{C}^\infty_c(M,\mathbb{R})\ ,$$ where $d\mu_g$ is the volume form associated to the Lorentzian metric $g$. One can understand the test functions $f$ as "component indices", just like we do in the case of the $2n$-dimensional symplectic (phase) space generated by $n$ positions $x_1,\ldots,x_n$ and $n$ momenta $p_1,\ldots,p_n$. Given any non-void open subset $O\subset M$ of the space-time manifold $M$, let $\tilde{\mathfrak{A}}(O)$ be the unital $*$-algebra generated by the Weyl unitaries $W_f$ as $f$ runs over the real-valued smooth functions compactly supported in $O$, once we endow such functionals with the following operations:

The above operations are then extended to general elements in the usual way, and they entail that the Weyl unitaries are worthy of their name, since we clearly have $W^*_f W_f=\mathbb{1}$. This $*$-algebra has non-trivial $*$-representations (namely, Fock representations associated to quasi-free states), hence it admits a minimal nontrivial C$*$-norm $\|\cdot\|$. The Weyl algebra $\mathfrak{A}(O)$ associated to $O$ is the C$*$-completion of $\tilde{\mathfrak{A}}(O)$ with respect to this C$*$-norm, and the correspondence $O\mapsto\mathfrak{A}(O)$ is obviously an isotonous net of C$*$-algebras. Moreover, due to the causal support of $\Delta_{m,g}$ entailed by the hyperbolicity of the Klein-Gordon operator, this net is also causal (i.e. elements localized in causally disjoint regions commute). Finally, uniqueness of retarded and advanced fundamental solutions for this operator guarantees that the isometry group of $(M,g)$ acts on this net as it should - namely, for any isometry $\psi$ of $(M,g)$, the action $$ \alpha_\psi(W_f)(\phi)=W_f(\psi^*\phi)=W_{\psi_*f}(\phi) $$ uniquely extends to *-isomorphisms satisfying $\alpha_\psi\circ\alpha_{\psi'}=\alpha_{\psi\circ\psi'}$ and $\alpha_\psi(\mathfrak{A}(O))=\mathfrak{A}(\psi(O))$. In the above formula, $\psi^*\phi=\phi\circ\psi$ is the pullback of the field configuration $\phi$, and $\psi_*f=f\circ\psi^{-1}$ is the pushforward of the test function $f$. To summarize, we have obtained a Haag-Kastler net of C$*$-algebras.

The Weyl algebra construction can be done abstractly for any real vector space (even infinite-dimensional) endowed with an antisymmetric bilinear form, thanks to B. Blackadar's universal C*-algebra construction using generators and relations ("Shape theory for C∗-algebras", Math. Scand. 56 (1985) 249–275). However, since you asked for a concrete correspondence, here is a more explicit description. In the case of (free) real scalar fields $\phi$ in a globally hyperbolic space-time $(M,g)$ subject to the Klein-Gordon equation $$ \Box_g\phi+m^2\phi=0\ ,$$ the Weyl unitaries are concretely given by the functionals $$ \mathscr{C}^\infty(M,\mathbb{R})\ni\phi\mapsto W_f(\phi)=e^{i\int_M f\phi\ d\mu_g}\ ,\quad f\in\mathscr{C}^\infty_c(M,\mathbb{R})\ ,$$ where $d\mu_g$ is the volume form associated to the Lorentzian metric $g$. One can understand the test functions $f$ as "component indices", just like we do in the case of the $2n$-dimensional symplectic (phase) space generated by $n$ positions $x_1,\ldots,x_n$ and $n$ momenta $p_1,\ldots,p_n$. Given any non-void open subset $O\subset M$ of the space-time manifold $M$, let $\tilde{\mathfrak{A}}(O)$ be the unital $*$-algebra generated by the Weyl unitaries $W_f$ as $f$ runs over the real-valued smooth functions compactly supported in $O$ (so we say that such $W_f$'s are localized in $O$), once we endow such functionals with the following operations:

The above operations are then extended to general elements in the usual way, and they entail that the Weyl unitaries are worthy of their name, since we clearly have $W^*_f W_f=\mathbb{1}$. This $*$-algebra has non-trivial $*$-representations (namely, Fock representations associated to quasi-free states), hence it admits a minimal nontrivial C$*$-norm $\|\cdot\|$. The Weyl algebra $\mathfrak{A}(O)$ associated to $O$ is the C$*$-completion of $\tilde{\mathfrak{A}}(O)$ with respect to this C$*$-norm, and the correspondence $O\mapsto\mathfrak{A}(O)$ is obviously an isotonous net of C$*$-algebras. Moreover, due to the causal support of $\Delta_{m,g}$ entailed by the hyperbolicity of the Klein-Gordon operator, this net is also causal (i.e. elements localized in causally disjoint regions commute). Finally, uniqueness of retarded and advanced fundamental solutions for this operator guarantees that the isometry group of $(M,g)$ acts on this net as it should - namely, for any isometry $\psi$ of $(M,g)$, the action $$ \alpha_\psi(W_f)(\phi)=W_f(\psi^*\phi)=W_{\psi_*f}(\phi) $$ uniquely extends to unit-preserving *-isomorphisms satisfying $$\alpha_\psi\circ\alpha_{\psi'}=\alpha_{\psi\circ\psi'}\ ,\quad\alpha_\psi(\mathfrak{A}(O))=\mathfrak{A}(\psi(O))\ .$$ In the above formula, $\psi^*\phi=\phi\circ\psi$ is the pullback of the field configuration $\phi$, and $\psi_*f=f\circ\psi^{-1}$ is the pushforward of the test function $f$. To summarize, we have obtained a Haag-Kastler net of C$*$-algebras.