One way a person could stumble on quadratic reciprocity while looking at theta functions is by trying to prove that Weil's construction of an adelic "Segal-Shale-Weil/oscillator representation" really is a representation, and really produces automorphic forms. This would lead a person to see that certain products of local characters must be Hecke characters... and that Poisson summation exactly proves this.
In somewhat more detail: given a local field $k$ (maybe not char 2), the "basic"local Segal-Shale-Weil repn is a repn of a two-fold cover $Mp_1(k)$ on Schwartz functions on $k$. For $K$ either a quadratic extension of $k$ or $k\oplus k$, the analogous repn of $Mp_1(k)$ descends to $SL_2(k)$, since the relevant cycle "splits". Without even knowing how to verify that splitting, one could still just try to directly assemble this local repn from a Bruhat decomposition and from the action of the standard unipotent radical, the standard Levi, and the Weyl element.
That is, with the rough idea that the Weyl element should be Fourier transform, up to sign..., that the Levi component should act by dilation, up to maybe a character and a norm to make the operator unitary, and that the unipotent radical should act by multiplication by a quadratic exponential... (which one might get from the repn of the Lie algebra of $SL_2(\mathbb R)$ on various function spaces on $\mathbb R$!!!)... one sees that for these pieces to fit together to make a repn, the Levi action is indeed not just dilation (with adjustment by norm), but twisted by the norm residue character attached to $K/k$. This gives a repn of $O(N_{K/k})\times SL_2(k)$ viewing the norm as a quadratic form on $K$.
When the local situation arises across all places from a number field extension $K/k$ (or function field, too, tho' eschewing char$=2$), one can try to apply this to the global pair $O(N_{K/k})\times SL_2(k)$ to make binary theta series and "special waveforms" by lifting Hecke characters from the adele group of $SO(K/k)$ to the adele group of $SL_2(k)$. This is the smallest interesting "theta correspondence", and the holomorphic case was known to Hecke, and the waveform case was Maass' thesis, though of course they wrote things classically and over $\mathbb Q$.
To a Schwartz function $\varphi$ on the adeles of $K$ attach a "theta kernel" by $\theta_\varphi(g,h)=\sum_{\lambda\in K} \big((g,h)\cdot \varphi\big)(x+\lambda)$ where $g,h$ are in the adelized $SL_2(k)$ and $O(N_{K/k})$. This is immediately invariant under the action of the rational points of $O(N_{K/k})$, but it is not obvious that it is invariant under $SL_2(k)$. But, as expected (!), one can verify that it is.
One point is that the character by which the Levi acts is a Hecke character. If not, the theta kernel will not be suitably invariant... but/and one uses Poisson summation and natural computations to prove that it is a Hecke character. Great! That is, one proves that the product of the local norm residue symbols is a Hecke character. From this, one proves the corresponding reciprocity for the quadratic Hilbert symbols, and then for quadratic symbols.
The argument is not as short as one might wish... but is fairly natural.
I did not quite see/find exactly that argument in Weil, despite looking again later, perhaps because of the style of his piece, but/and I'm sure he would say it is implicit, and one sees that it must be.
After I went through the above story first-hand years ago, I wrote up the spin-off that proves quadratic reciprocity over global fields not char zero... a recently-slightly-revised version is at http://www.math.umn.edu/~garrett/m/v/quad_rec_02.pdf
