I really hope everyone either gets their game or a full refund.
To clarify something, a for-profit corporation has shareholders. The shareholders are the owners and are entitled to the distribution of the profits of the company in proportion to the percentage of their shares of stock.
A non-profit corporation does not have shareholders. No one is entitled to the distribution of the profits of the company (basically, they are recycled into the company). Upon the dissolution of the non-profit corporation, the articles of incorporation will provide how the assets of the company are distributed, usually to some general purpose that the non-profit corporation was pursuing or some specific similar organization. A non-profit corporation is not automatically a "tax-exempt" organization.
To be a tax exempt organization, the non-profit corporation, which is organized through the State, must then apply to the Internal Revenue Service and meet the qualifications for tax exemption. The one most people are familiar with is 501(c)(3) (the applicable section of the Internal Revenue Code)- tax exempt as a "charitable" organization.
Just because a company is non-profit doesn't mean it can't make money, pay salaries, bonuses, buy and hold assets etc. it just means that after all the expenses are taken against revenue, no profits are distributed to the shareholders (because it doesn't have them). If it has not achieved tax exempt status, it would still be subject to tax on its profits, if any (usually the corporation would just buy equipment or something to avoid profits - why pay tax when you can use that money to further the company's purpose).
That all said
I don't know what they named the company (it wouldn't necessarily be Skit-B pinball, that could just be a doing business as), but I searched Michigan's business entities with the keywords pinball, Kulek and Skit-B and didn't find any registered company that I would identify as this organization. The link is above.
Just reading the posts of people who are in on the game, I really hope this works out for you.