I found an answer to my own question from Ranjan Roy's book "Elliptic and Modular Functions from Gauss to Dedekind to Hecke".
On page 293, Roy says the $q$-product of the $\Delta$-function is not due directly to Jacobi, but to Klein. In Klein's 1878/1879 paper Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades (pp. 13-75 in Volume 3 of his collected works here), $\Delta$ is defined on the top of the 3rd page as a discriminant and in equation (18) on the ninth page you can see the $q$-product for the 12th root of $\Delta(\omega_1,\omega_2) := \Delta(\omega)$, where $\omega = \omega_1/\omega_2$ and Klein's $q^2$ is $e^{2\pi i\omega}$. Taking $12$th powers of both sides of the equation and setting $\omega_2 = 1$ yields the usual product for $\Delta$.
The second part of Klein's equation (17) expresses the $12$th root of $\Delta$ in terms of a product of three functions whose cube he expresses just above his equation (18) as a $q$-product that is taken from equation (2) on p. 89 of Jacobi's Fundamenta Nova here. That's the basis for attributing the $q$-product of the $\Delta$-function to Jacobi. Since Klein's work relied on Jacobi, he needed knowledge of elliptic functions.
Roy says earlier in his book, on p. 99, that Weierstrass used his sigma function to derive the $q$-product of $\Delta$ in his lectures on elliptic functions in 1862-1863: see equation 12 on p. 164 of the 5th volume of Weierstrass collected works here, where $G$ is $\Delta$ up to powers of $2$.
Hurwitz, who was a PhD student of Klein, was the first to develop modular functions without needing prior knowledge of elliptic functions (or theta functions) of any kind, and in particular he proved the product for $\Delta$ without needing elliptic functions. This is the approach to modular functions used in introductory book today, based on Eisenstein series as a starting point.