Just as Ryan Budney pointed out, instead of ambient isotopy one may consider a weaker equivalence relation on spatial graphs, namely the one generated by isotopy and IH-moves (also known as Whitehead moves). With this definition of equivalence, two knotted graphs are equivalent if and only if they admit isotopic regular neighbourhoods. This equivalence relation has been already considered, for example, by Kinoshita in 1958, and it was named ''neighbourhood equivalence'' for obvious reasons.
Of course, the study of graphs up to neighbourhood equivalence reduces to the study of knotted handlebodies. There exist several invariants of knotted handlebodies. Among them, I have recently become interested in the quandle coloring invariants defined by Ishii in his paper
Moves and invariants for knotted handlebodies
Algebraic & Geometric Topology 8 (2008) 1403–1418
In a joint paper with R. Benedetti
"Levels of knotting of spatial handlebodies"
we have exploited (among other things, like the Alexander invariants of the complement) these quandle coloring invariants in order to distinguish different levels of knotting for handlebodies.
Just as in the case of knot theory, a good invariant for a knotted handlebody is its complement. However, while Gordon-Luecke's Theorem ensures that a knot is determined by its complement, there exist inequivalent handlebodies whose complements are homeomorphic
(this is one of the reasons why I would compare the theory of knotted handlebodies of genus g with the theory of links with g components, rather than with knot theory). On the other hand, Kent and Souto exhibited here
a spatial handlebody whose complement admits a unique embedding in the 3-sphere up to isotopy. Also observe that, due to Fox's reimbedding Theorem, every compact submanifold of the 3-sphere admits a reimbedding as the complement of a finite union of handlebodies in the 3-sphere itself. Therefore, a complete understanding of knotted handlebodies should provide an understanding of spatial domains in general.