Wlazły spent most of his career in Skra Bełchatów, the club with which he won 9 league titles and multiple medals in international competitions.

With PGE Skra, he won a bronze medal at the 2007–08 CEV Champions League, and was also individually awarded as the Best spiker of the tournament.

His club played in the Final four of the 2014–15 CEV Champions League, but lost in the semifinal to Asseco Resovia, and in a match for third place to Berlin Recycling Volleys, and did not achieve any medal.

In May 2015, he signed a contract with Qatari club Al Arabi Doha for the duration of the Emir Cup (only 2 matches).

On 8 April 2014, Wlazły was appointed to the Polish national team after 4 years of absence, by the head coach Stéphane Antiga.

[12] On 27 October 2014, he received a state award granted by the Polish President Bronisław Komorowski – Officer's Cross of Polonia Restituta for outstanding sports achievements, and promotion of Poland in the world.