The post currently has 16 upvotes. So there is some support for this proposal.
There is no indication that any group of paid employees or volunteer employees of Stack Exchange (SE) are excluded by the "National Labor Relations Act"
(3) The term "employee" shall include any employee, and shall not be limited to the employees of a particular employer, unless the Act [this subchapter] explicitly states otherwise,
In fact most of us would fall in the category of "professional employee", it seems self evident that all Moderators fall into this category.
(12) The term "professional employee" means--
(a) any employee engaged in work (i) predominantly intellectual and varied in character as opposed to routine mental, manual, mechanical, or physical work; (ii) involving the consistent exercise of discretion and judgment in its performance; (iii) of such a character that the output produced or the result accomplished cannot be standardized in relation to a given period of time; (iv) requiring knowledge of an advanced type in a field of science or learning customarily acquired by a prolonged course of specialized intellectual instruction and study in an institution of higher learning or a hospital, as distinguished from a general academic education or from an apprenticeship or from training in the performance of routine mental, manual, or physical processes; or
(b) any employee, who (i) has completed the courses of specialized intellectual instruction and study described in clause (iv) of paragraph (a), and (ii) is performing related work under the supervision of a professional person to qualify himself to become a professional employee as defined in paragraph (a).
We have the right to self organize
RIGHTS OF EMPLOYEES
Sec. 7. [§ 157.] Employees shall have the right to self-organization, to form, join, or assist labor organizations, to bargain collectively through representatives of their own choosing, and to engage in other concerted activities for the purpose of collective bargaining or other mutual aid or protection, and shall also have the right to refrain from any or all of such activities except to the extent that such right may be affected by an agreement requiring membership in a labor organization as a condition of employment as authorized in section 8(a)(3) [section 158(a)(3) of this title].
Our employer (SE) is prohibited from interfering with our attempts to organize
UNFAIR LABOR PRACTICES
Sec. 8. [§ 158.] (a) [Unfair labor practices by employer] It shall be an unfair labor practice for an employer--
(1) to interfere with, restrain, or coerce employees in the exercise of the rights guaranteed in section 7 [section 157 of this title];
But it does appear that SE is also prohibited from providing support, so an Area 51 proposal to create a site, is probably not feasible. But general posts about the union, would seem to be allowed.
(2) to dominate or interfere with the formation or administration of any labor organization or contribute financial or other support to it: Provided, That subject to rules and regulations made and published by the Board pursuant to section 6 [section 156 of this title], an employer shall not be prohibited from permitting employees to confer with him during working hours without loss of time or pay;
I am in no way an expert on this topic. If you can expand on or clarify any points, please post a new answer.
I guess the next step would be to find a meeting hall.