Accessing research databases with company computer? Are there any legal (or other) limitations in accessing research databases from a computer at my workplace? 

As a part-time student I have full access to resources provided by my university and was wondering whether I should stop using company's laptop for research purposes.  

I'm based in UK, if that matters.
You should check with your library policies, but use of academic library (and other) resources for work outside an academic educational/research context is very likely against policy and may violate the licenses by which the university has obtained access to material (i.e., the library has obtained a license for the students and staff of the university to access the material for noncommercial purposes related to academic research and education; they likely have not obtained a license for those students and staff to access the material for commercial purposes outside the university).

Using your company's laptop might be a bit of a grey area since you are transferring materials to the property owned by the company; if you are using the company laptop for academic purposes, you might be in violation of company policies but it has nothing to do with the library access. Whether simply accessing library materials from a commercial source if you are not using those resources to do non-licensed work (i.e., to use the access for commercial purposes) seems like a question only a lawyer could answer, and you won't get legal advice here.