Lazar was ally of Tvrtko. He did not recognise Tvrtko as King of Serbia. None of Serbian lords did it. Nor they did acknowledge legitimate Serbian King Marko Kraljević. Lazar considered himself Lord of Serbs and was lobbing to be elected as King or even Tsar. Epic poetry did consider him Tsar. His wife was from colateral branch of Nemanjići, his father Pribac was Serbian Prime Minister. Lazar and his sons-in-law controled most of Serbia, so he was not willing to acknowledge anyone abowe him.
Battle of Kosovo was fought precisely because he did not wish to be somebodies vasal.
But on other hand he and Tvrtko were close allies. Tvrtko was highly relaistic and know that dispite his blood connection with Nemanjići (Great-grandson of King Dragutin), he controled small parts of Serbia. Newerttheless he and Lazar allways acted together.