Replicants certainly do have emotions - Roy Batty exhibits anger and ultimately sadness; several of them exhibit fear at being discovered and retired.
However, they are relatively underdeveloped, emotionally speaking. Replicants are physically matured and "born" at full adult growth - they don't go through a childhood to socially and emotionally mature like regular humans. The VK test is not aimed at detecting a lack of emotion - it is designed to detect inappropriate response to emotionally stimulating scenarios.
It seems that this emotionally stunting is known to be a problem - one of the reasons for the programmed short life-spans of the Nexus-6 models is because they become more "emotionally erratic" - Roy, Pris, Leon, and Zora are all close to the end of their life.
Tyrell Corporation were working on improving this - by programming in childhood memories - thereby providing some emotional maturity to the fresh new replicant.