I think you're looking for the work of Tomio Kubota.
- The square of the theta function is a modular form. For a while, and still today, the theta function itself is sometimes considered a modular form "of half-integral weight".
However, mostly through work of André Weil, it was then understood that $\operatorname{SL}_2$ of the adèles of a number field admits a degree 2 central extension, the metaplectic extension. Other answers have mentioned this. The theta function is then most elegantly understood as an automorphic form not for $\operatorname{SL}_2$ itself, but for this central extension.
People like Calvis C. Moore, Matsumoto, Steinberg and Kubota have then discovered that the cocycle describing this central extension (recall that central extensions of groups are classified by cohomology classes in $H^2$ of group cohomology) permits a simple expression in terms of the classical Hilbert symbol. You can find out more about this if you google for "Kubota cocycle" formula (or just look it up in the book of Kubota cited below).
Once you phrase quadratic reciprocity as Hilbert reciprocity for Q, quadratic reciprocity then becomes equivalent to the statement that the central extension splits when $\operatorname{SL}_2$ is pulled back along the diagonal ring map $\mathbb{Q}^{\times} \hookrightarrow $ adèles.
But if this holds, it should be reflected by a property of the automorphic form that is the theta series. And indeed, unravelling this leads you exactly to the observation you're making about theta reciprocity.
- The same people then discovered that the adèles of any number field possess a unique topological central extension, then also called metaplectic extension. In general it is a central extension of $\operatorname{SL}_2$ of degree equal to the number of roots of unity in the number field
(so for Q, where +1 and -1 are the only roots of unity, so this is a degree 2 covering)
There is a corresponding theory of modular forms for these extensions. They are in general called metaplectic forms. There is a very important foundational paper by Kazhdan-Patterson about this (Metaplectic Forms, Publ IHES).(beware of a sign mistake, explained in follow-up literature, or ask Daniel Bump or Ben Brubaker about this)
It is very hard to get much explicit control for this type of metaplectic form. Anyway, Kubota's little book https://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/84907/1/Lectures_in_Mathematics_2.pdf (Automorphic Forms and Reciprocity Law for Number Fields) shows that nonzero metaplectic forms really exist, but the proof is less explicit than you might like. Patterson has some papers about explicit formulae for number fields with a primitive third root of unity. It's all very intimidating stuff, and the whole field is notorious for how easily sign (or worse) mistakes slip into computations. It's the kind of maths which can only be done by super-careful people.
Anyway. The much-desired theta reciprocity law for general (degree n) metaplectic theta series for arbitrary number fields with a primitive n-th root of unity has still not been found (as of April 2024).But it should exist. For a general number field, the Hilbert reciprocity law should follow. This would be a spectacular result, at least if you manage to bypass using class field theory as input.
Sadly, not very many mathematicians are still active in this quest. Some of the leading figures are retired or have passed away. Still, it would be absolutely spectacular.
At present, it's difficult even to just run experiments, for our ability to explicitly do computations with metaplectic forms is far too limited. Contact people like Brubaker, Weissman or perhaps Toshiaki Suzuki for more on this. Stay strong!
- I've talked a lot about $\operatorname{SL}_2$, but you can do all this for more general groups, look at work of Weissman or the famous paper Brylinski-Deligne on central extensions by $K_2$. However, if you only want the connection between reciprocity laws and theta series, the $\operatorname{SL}_2$ situation should be enough for all you want. In this setting, the central extension is essentially unique, so you can ignore all of the Brylinski-Deligne structure theory.