It is somewhat surprising (to me) that what to me seems the *simplest nontrivial example of theorems exactly fitting the question in the OP has not yet been mentioned in this thread*: the embeddings of the [Möbius ladders][1], which are finite simple undirected graphs, into $\mathbb{R}^3$.
This is an answer to
> whether there is a "knot theory" for graphs, i.e. the study of (topological properties of) embeddings of graphs into R^3 or S^3.
If yes, can anyone show me any reference?
at least in the sense that in the very interesting article [Erica Flapan: The Symmetries of the Möbius Ladder. Math. Ann. 283, 271-283 (1989)][2], which I think, could be given a fruitful revival from a point of view of *constructive mathematics* (e.g., how much of Flapan's proofs/theorems can be done constructively?), the following was done, inter alia (there is more in Flapan's paper):
* in Section 1 of loc. cit. a proof is given that there exists a graph $G$ and an automorphism $\alpha$ of $G$ as an abstract graph, such that there *does not exist* any embedding of (the geometric realization $\lvert G\rvert$ of ) $G$ into $\mathbb{R}^3$ such that $\alpha$ could be realized by an element of the group of leaving-$\lvert G\rvert$-invariant-as-a-point-set diffeomorphisms of $\mathbb{R}^3$. Note that in this rendition I rendered the author's "group of homeomorphisms of $G$ up to isotopy" into what to me evidently seems equivalent and clearer "automorphism of $G$ as an abstract graph". The example graph used by the author is $G:=$complete graph with six vertices.
* in Section 2 of loc. cit., first a proof in classical logic is given that for any embedding $\eta\colon M_r\to S^3$ of the $r$-rung Möbius ladder $M_r$ into the $3$-sphere $\mathbb{S}^3$, any orientation-reversing diffeomorphism $\varphi\colon S^3\to S^3$ has the property that *if* it fixes<sup>1</sup> $\mathrm{im}(\eta)$ as a pointset, then $\varphi$ does *not* fix at least $r-2$ of the $r$ rungs of $M_r$. This is, *partly*, an interesting counterfactural (since the author later gives a proof that for an **odd** number of rungs, such $\varphi$ are impossible): roughly, *if* there is a symmetry of the Euclidean-space-embedded Möbius ladder, *then* it must needs jumble almost all the ladder's rungs. This implication is then put to use to give a proof that for an **odd** number of $r$ungs, such a $\varphi$ is impossible. Roughly: you cannot reverse the orientation of an embedded odd-rung-number Möbius ladder *graph* by a Euclidean symmetry. The author then gives an example that for **even** number of rungs, such isometries *do* exist. I perceived this to be a result which is very relevant to the OP; it in particular is an interaction between a combinatorial property of the abstract graph (*number of rungs*) and a [concept studied in knot theory][3]: slightly vaguely, one could say: **Flapan gave a proof that even-rung Möbius ladders are amphichiral, while odd-rung Möbius ladders are non-amphichiral.** The smallest example of the latter ladder is the embedding represented by the illustration
[![enter image description here][4]][4]
on p. 272 of loc. cit. about which Flapan says that it is *non-amphichiral*, i.e. there does not exist any orientation-reversing self-diffeomorphism of $\mathbb{R}^3$ which whould fix it as a point set.
The author on p. 272 of loc. cit. writes
> The property of topological achirality for graphs is analogous to the property of amphicheirality for knots.
* in Section 3 of loc. cit., the focus shifts from *(non-)existence* of orientation-*reversing* Euclidean symmetries of embedded Möbius ladders, to *properties* of orientation-*preserving* such symmetries. The emphasis is on a difference between embeddings into $\mathbb{R}^3$ and embeddings into $S^3$.
This difference is illustrated by the author using the following illustration on p. 278 of loc. cit.
[![enter image description here][5]][5]
which is a an example of what the OP is asking for: this meant to represent a non-knot embedded into $S^3$, namely the abstract undirected simple graph $M_3$, the three-rung Möbius ladder.
**Remarks.**
* I realized that there is an issue with the rendition I gave above: in loc. cit. the self-diffeomorphism of $S^3$ which is shown to be impossible is assumed to map what is called in loc.cit. the 'loop' $K$ of the geometrically Möbius ladder to itself, so strictly speaking, the above presentation claims that loc. cit. proved *more* than loc. cit. claims it proves. However, personally, I do not understand the hypothesis $h(K)=K$ in Theorem 2 of loc. cit. in the sense that to me it seems *evident* that the hypothesis $h(M_n)=M_n$ in Theorem 2 *implies* $h(K)=K$ (because of $h(M_n)=M_n$ $\Rightarrow$ the abstract graph-homomorphism $a$ defined by $h$ is a graph-automorphism of $M_n$ $\Rightarrow$ $a$ maps the abstract graph underlying the 'loop' $K$ to itself $\Rightarrow$ $h(K)=K$, the latter implication because $h$ is assumed to satisfy $h(M_n)=M_n$) so it seems it can simply be left out. This is what the footnote <sup>1</sup> is about.
<sup>1</sup> Note that in loc. cit. there is a slightly *stronger* hypothesis than mere fixing the image of the embedding in its entirety. This hypothesis, to me, seems superfluous, so I think loc. cit. gives a proof of what is claimed in this thread.
[1]: https://en.wikipedia.org/wiki/M%C3%B6bius_ladder
[2]: https://link.springer.com/article/10.1007%2FBF01446435
[3]: https://en.wikipedia.org/wiki/Chiral_knot
[4]: https://i.sstatic.net/LYgtc.png
[5]: https://i.sstatic.net/D9LOa.png