It is fairly straightforward to adapt standard Gröbner basis techniques to such algebras, e.g. see the [paper \[1\]][1]. See also the [paper \[0\]][0] which applies such algorithms to the problem at hand.

[0]: https://doi.org/10.1007/11870814_12
[[0]] Jesus Gago-Vargas; Isabel Hartillo-Hermoso; Jose Marya Ucha-Enryquez  
Algorithmic Invariants for Alexander Modules. LNCS 4194, 149-154  
https://link.springer.com/chapter/10.1007/11870814_12

Abstract. Let G be a group given by generators and relations. It is possible
to compute a presentation matrix of a module over a ring through
Fox's differential calculus. We show how to use Gröbner bases as an algorithmic
tool to compare the chains of elementary ideals defined by the
matrix. We apply this technique to classical examples of groups and to
compute the elementary ideals of Alexander matrix of knots up to 11
crossings with the same Alexander polynomial.


[1]: https://doi.org/10.1007/s002000050108
[[1]] Franz Pauer, Andreas Unterkircher.  
Gröbner Bases for Ideals in Laurent Polynomial Rings and their Application to Systems of Difference Equations.  
AAECC 9, 271-291 (1999)  
https://link.springer.com/article/10.1007/s002000050108

*Abstract.* We develop a basic theory of Gröbner bases for ideals in the algebra
of Laurent polynomials (and, more generally, in its monomial subalgebras). For
this we have to generalize the notion of term order. The theory is applied to
systems of linear partial difference equations (with constant coefficients) on
${\mathbb Z}^n$. Furthermore, we present a method to compute the intersection of an ideal
in the algebra of Laurent polynomials with the subalgebra of all polynomials.