Does this Vigilance Mechanism apply to the Private Limited companies ?

Answer is Yes, it does apply in some cases

Section 177 of the Companies Act 2013

(9) Every listed company or such class or classes of companies, as may be prescribed, shall establish a vigil mechanism for directors and employees to report genuine concerns in such manner as may be prescribed

(10) The vigil mechanism under sub-section (9) shall provide for adequate safeguards against victimisation of persons who use such mechanism and make provision for direct access to the chairperson of the Audit Committee in appropriate or exceptional cases:

Provided that the details of establishment of such mechanism shall be disclosed by the company on its website, if any, and in the Board’s report.

Section 178 of the Companies Act 2013

Section 178 (8) In case of any contravention of the provisions of section 177 and this section, the company shall be liable to a penalty of five lakh rupees and every officer of the company who is in default shall be liable to a penalty of one lakh rupees.

Rule 7 of the Companies (Meetings of Board and its Powers) Rules,2014

Under Companies Act

7. Establishment of vigil mechanism.-

(1) Every listed company and the companies belonging to the following class or classes shall establish a vigil mechanism for their directors and employees to report their genuine concerns or grievances-

(a) the Companies which accept deposits from the public;

(b) the Companies which have borrowed money from banks and public financial institutions in excess of fifty crore rupees.

(2) The companies which are required to constitute an audit committee shall oversee the vigil mechanism through the committee and if any of the members of the committee have a conflict of interest in a given case, they should recuse themselves and the others on the committee would deal with the matter on hand.

(3) In case of other companies, the Board of directors shall nominate a director to play the role of audit committee for the purpose of vigil mechanism to whom other directors and employees may report their concerns.
(4) The vigil mechanism shall provide for adequate safeguards against victimisation of employees and directors who avail of the vigil mechanism and also provide for direct access to the Chairperson of the Audit Committee or the director nominated to play the role of Audit Committee, as the case may be, in exceptional cases.

(5) In case of repeated frivolous complaints being filed by a director or an employee, the audit committee or the director nominated to play the role of audit committee may take suitable action against the concerned director or employee including reprimand.