There is a description of the appropriate Groebner basis algorithm in this book:

Franz Pauer, Andreas Unterkircher.
[Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations.][1]
AAECC 9, 271–291 (1999)

I've implemented it in the single-variable case (in the software Regina) and I've been meaning to implement it in the multi-variable case as well.  But I usually get too sad to finish – when I look at how inefficient the algorithm is.   Some day I'll have it fully implemented in Regina.

The book is quite well written.  I find it easy to read. 

  [1]: https://doi.org/10.1007/s002000050108