I am fairly sure the following lemma is folklore, but for some weird reason I am unable to track it down in the literature.
Suppose $i:\mathcal{C}\hookrightarrow \mathcal{D}$ is a full reflective subcategory with reflector $L:\mathcal{D}\rightarrow \mathcal{C}$. Suppose further that $\mathcal{D}$ has a (not necessarily symmetric, not necessarily closed) monoidal structure $(\otimes,1,\alpha,\lambda,\rho)$.
Assume that this tensor product preserves local morphisms, ie. for morphisms $f:x\rightarrow y, g:u\rightarrow v$ in $\mathcal{D}$ for which $L(f),L(g)$ become isomorphisms in $\mathcal{C}$ also the monoidal product $f\otimes g$ is local in that $L(f\otimes g)$ is an isomorphism in $\mathcal{C}$. Then $\mathcal{C}$ admits a unique monoidal structure, with tensor product $c \boxtimes c' := L(i(c)\otimes i(c'))$ and monoidal unit $L(1)$, such that $L$ is a strong monoidal functor.
The idea is to first realize that reflective subcategories are very well behaved categorical localizations at the class of local morphisms. Now, if $\mathcal{C}\subseteq \mathcal{D}$ is reflective, so is $\mathcal{C\times C}\subseteq \mathcal{D\times D}$. The condition that the tensor preserves local morphisms is precisely what one needs to obtain an induced functor on the localizations $\boxtimes:\mathcal{C\times C} \rightarrow \mathcal{C}$. Using the fact that this induced functor is a Kan-extension one can then verify that this is product of a monoidal structure on $\mathcal{C}$ as claimed.
I looked into Borceux's Handbooks of Categorical Algebra, MacLane's Categories for the Working Mathematician, Aguiar&Mahajan's Monoidal Functors, Species and Hopf Algebras and Riehl's Category Theory in Context with no avail.
If you search for monoidal reflection on the internet, Day's reflection theorem and Day's Note on Monoidal Localization seem to be the only immediately related resources. However, if I am not mistaken, he uses the monoidal closure, which I cannot.
So, if you know a reference, please let me know. As always, thank you very much for your time.
PS: If I am not mistaken, the $\infty$-categorical analogue of this statement is proposition 2.2.1.9 in Lurie's Higher Algebra.