Any local public entity located in this state specified in paragraph (3) of subdivision (a) of Section 135 or Indian tribe specified in paragraph (6) of subdivision (a) of Section 135 may elect to become an employer subject to Part 2 (commencing with Section 2601) of this division with respect to all its employees, including those with civil service or tenure positions, and may file its written election with the director. That election may be made on its own motion by the appropriate governing board of the local public entity or Indian tribe making the election, or may be made by the governing board pursuant to a petition signed by a majority of the employees (including those with civil service or tenure positions) requesting the governing board to file an election with the director. Upon the filing of an election, the filing local public entity or Indian tribe shall, upon approval by the director, become an employer subject to Part 2 (commencing with Section 2601) to the same extent as other employers, and services performed by its employees, including those with civil service or tenure positions, shall constitute employment subject to that part. Beginning at that time, it shall withhold from the wages of employees the contributions required for unemployment compensation disability benefits.