Are there moves between Reidemeister moves?

In some strict sense I think the answer to your question is no, there are likely no finite collection of 2-cells doing what you want. If you were to ask the more natural question where you're looking for a 2-complex whose inclusion into $Emb(S^1,\mathbb R^3)$ is an isomorphism on $\pi_1$ (component-by-component) the answer I'm near-certain is yes (via Thom-Mather singularity theory).

For example, here is a non-trivial loop in the space of knots which you could imagine as a loop in your Reidemeister graph once you refine things suitably. This loop isn't a problem if you only want the map (2-complex) $\to Emb(S^1,\mathbb R^3)$ to be an isomorphism on $\pi_1$. But for the complex you want, these loops are a problem, as they're very much global things and can't be described readily in terms of local diagram moves.

The loop described in this picture can be done for any combination of summands -- as long as the summand knots are non-trivial this is a non-trivial loop. So how are you going to construct a finite collection of 2-cells that kill off all these loops?

Take a look at page 180 of Low dimensional topology by Tomasz Mrowka, Peter Steven Ozsvát, (following Ben Webster's comment about Movie Moves elucidated by Baez and Langford and their 30 basic movie moves, and by Carter and Saito who describe a 31st basic movie move.) A movie move is a sequence of frames of a braid (or subregion of a knot, I suppose).

Carter and Saito have a theorem that

two movies represent the same tangle cobordism iff they can be related by a sequence of movie moves

If you take the subset of Movie Moves where each movie is a composition of a sequence of Reidemeister moves, it seems like that would be similar or equivalent to what you are calling "Higher Reidemeister moves." Am I understanding you correctly?

I would point you out to the appropriate page on that wiki o' info, but "Movie moves" does not even show up on their search page.