They want State House to exclusively engage all Coast leaders, including Senators, MPs and Assembly Leadership, before any other sitting.
Thoya expressed dismay at what he perceived as the national government's neglect of coastal leaders' concerns about muguka.
Lamu joins Mombasa, Kilifi, and Taita Taveta as the fourth county to enforce such a ban, following announcements made by the coastal counties last month.
The legislator argued that determining whether a substance is a drug or not is a matter of scientific testing, not parliamentary debate.
A study led by Kenya's well-known psychiatrist, Prof. Lukoye Atwoli, along with 13 other experts, reveals that people who chew khat often experience psychotic symptoms such as hallucinations, delusions, and disturbed thoughts.
The governor reported the seizure and destruction of 85 bags of muguka in the operation.
The President says he has engaged governors Abdullswamad Sherrif Nassir (Mombasa), Gideon Mung'aro (Kilifi) and Andrew Mwadime (Taita Taveta) and that they have agreed to a meeting.
The counties' governors spoke after President William Ruto's directive on Monday, nullifying a ban by Mombasa, Kilfi, and Tana River counties.