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 for rung-numbers $\geq 4$, each odd-rung Möbius ladder is non-amphichiral, while each even-rung Möbius ladder *is* amphichiral.** The smallest example of the latter ladders is the embedding represented by the illustration
[![enter image description here][4]][4]
on p. 272 of loc. cit., when thought of to *have precisely four rungs*, about which loc. cit. says that it *is* *amphichiral*, while if the picture is thought of to represent a Möbius ladder with *five* rungs, then loc. cit. says that there does not exist any orientation-reversing self-diffeomorphismof $\mathbb{R}^3$ which would fix it as a point set.
(The latter is a precise and usual statement of the matter. A more vague alternative statement one often encountered is 'It is not equivalent to its mirror-image.' wherein 'mirror-image' is either left undefined, or is defined via orientation-reversing self-homeomorphism of $\mathbb{R}^3$, whereupon the alternative statement becomes at least not simpler.)
The amphichirality of the even-rung Möbius ladders is easy to see, the substance of Flapan's results is non-amphichirality of the odd-rung Möbius ladders with at least 5 rungs.
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.
Furthermore, one should mention that there is a very recent preprint
> Erica Flapan, Thomas Mattman, Blake Mellor, Ramin Naimi, Ryo Nikkuni: Recent Developments in Spatial Graph Theory. arXiv:1602.08122v2 [math.GT]
which is relevant to the OP. Therein, more results on embedded Möbius ladders are summarized, in particular the new paper (published after the OP)
> E. Flapan and E. Davie Lawrence, Topological symmetry groups of Möbius ladders, J. Knot Theory Ramifications, vol. 23, no. 14, (2014)
**Remarks.**
<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 *is essential in the special case $r=3$, yet can be left out for all $r\geq 4$*, as loc. cit. itself points out on p. 272.
<sup>2</sup> The rung-number $r=3$ is an exceptional case. The $3$-rung Möbius ladder is isomorphic to the complete graph $K^{3,3}$, and its geometric realization in $\mathbb{R}^3$ happens to *be* amphichiral: it is evident that the geometric graph $G$ represented by the illustration
[![enter image description here][7]][7]
in loc. cit. is isotopic to its 'mirror-image'.
[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/Cpx1P.png
[5]: https://i.sstatic.net/D9LOa.png
[6]: https://en.wikipedia.org/wiki/Wagner_graph
[7]: https://i.sstatic.net/vF0rv.png