Yes, there's plenty of work on this.  First of all, you have to define the notion of equivalence that you are interested in.  Usually people only care about the graph up to handle-slide (turning the subject into the subject of knotted handlebodies), so you can assume the graph is tri-valent.   But you could go further to study graphs up to isotopy and there's work on that too.   Much of the technology to study knots translates to studying knotted graphs. Some references:

http://katlas.org/drorbn/index.php?title=The_Alexander_Polynomial_of_a_Knotted_Trivalent_Graph

http://katlas.org/drorbn/index.php?title=The_Kontsevich_Integral_for_Knotted_Trivalent_Graphs

http://ldtopology.wordpress.com/2009/10/29/which-knotted-objects-are-worthy-of-study/

http://www.ms.unimelb.edu.au/~snap/

[Link](https://projecteuclid.org/journals/experimental-mathematics/volume-19/issue-2/Hyperbolic-Graphs-of-Small-Complexity/em/1276784791.full)

The last two references are rather nice as they show that much the same way hyperbolic geometry "dominates" traditional knot theory, it plays a similar role in the study of knotted trivalent graphs.  In this case orbifolds play a more prominent role.