Khovanov homology usualy has torsion. But there are also different versions of Khovanov homology. Is there a Khovanov homology theory that naturaly does not produce torsion? A followup question: can I just work over vector spaces, instead of $\mathbb{Z}$modules if I am only interested in the free part?

4$\begingroup$ The answer to your second question is yes: you can build the Khovanov chain complex using vector spaces over a field such as $\mathbb Q$, which only sees the free part of Khovanov homology. $\endgroup$ – Arun Debray Mar 28 '18 at 20:19
Odd Khovanov homology (arXiv.0710.4300) by Ozsvath, Rasmussen, and Szabo is a version of Khovanov homology that typically (but not always) has less torsion than the standard Khovanov homology. For example, the odd Khovanov homology of an alternating link has no torsion, while the standard Khovanov homology of an alternating link usually has lots of torsion of order two. As an aside, a forthcoming result of Shumakovitch states that the only torsion in the Khovanov homology of an alternating link is of order two.
However, odd Khovanov homology still has plenty of torsion, and the relationship between the torsion in odd Khovanov homology and standard Khovanov homology is not completely understood. The stateofthe art in this comparison is a paper by Shumakovitch (arXiv.1101.5607).
The comment above is correct that one can simply work with coefficients some field (like $\mathbb{Q}$ or $\mathbb{F}_p$) in standard Khovanov homology, and then you will not have to worry about torsion. Specializing to a particular field for the coefficients often results in a weaker invariant than working with $\mathbb{Z}$ coefficients.