The general theorem that Fourier’s $\mathscr F:L^2(G)\to L^2(\hat G)$ maps convolution to product and vice versa is in Weil (1940, p. 113), for any locally compact abelian $G$ with dual $\smash{\hat G}$. But the special cases of the real line, the circle group (with $(f,g,h)(t)=$ $\sum (a_n,b_n,c_n)e^{int}$), and the integers:
$$
\begin{align}
&\mathbf{(R)}\qquad h(t)=\textstyle\int_{-\infty}^\infty f(s)g(t-s)\,ds &&\Rightarrow &&\textstyle\mathscr F(h)=\mathscr F(f)\mathscr F(g)\\
&\mathbf{(T)}\qquad h(t)=\smash[t]{\textstyle\int_0^{2\pi} f(s)g(t-s)\,ds} &&\Rightarrow &&c_n=a_nb_n\\
&\mathbf{(Z)}\qquad h(t)=f(t)g(t) &&\Rightarrow &&\textstyle c_n=\sum_k a_kb_{n-k}\\
\end{align}
$$
were known earlier:

Dieudonné (1981, p. 195) and Mackey (1980, p. 628) attribute $\mathbf{(R)}$ to **Lyapunov** (1900, 1901?); it’s also in Hausdorff (1901, p. 169), Poincaré (1912, p. 207), Lévy (1925, p. 184; 1928, p. 79), etc. (They state results in terms of added independent random variables, not repeating the step that $h(t)dt$ is the image of the product measure $f(s)ds\times g(t)dt$ under addition $\mathbf{R\times R\to R}$.)

Kahane and Lemarié-Rieusset (1995, p. 19) attribute $\mathbf{(T)}$ to **Fourier** (1822, p. 259) where it is rather implicit; it’s explicit in Young (1912, p. 32).

Burkhardt (1901, p. 84; 1914, p. 947) finds $\mathbf{(Z)}$ in Cauchy (1844, p. 1125) and earlier in **Euler** (1760), for both trigonometric polynomials (“product-to-sum”, pp. 176-186) and some series (p. 200); it’s also in Pringsheim (1886, p. 158), Hurwitz (1902, p. 369), Lebesgue (1906, §52), etc.

(And indeed, the history in Domínguez (2015, p. 46) cheerfully ignores all of the above.)

**Added:** Adams (2009) contains translations of Lyapunov’s (1900, 1901). This should be consulted, as e.g. Fischer (2011, p. 201) contradicts Dieudonné and Mackey by saying: “Lyapunov never used general concepts such as inversion formula or correspondence between convolution of distributions and products of characteristic functions” (and I am indeed not really seeing $\mathbf{(R)}$ in (1900)).