Normally there wouldn't be a single consolidated closing. Each purchase of equity in connection with the offering would have a separate closing.
The scope of the equity offering as a whole would be defined in the offering document. It wouldn't be unusual to have the proceeds of funds wired in exchange for stock to be held in escrow until a threshold is met, with the funds wired returned if the amount necessary to proceed with the business project it is financing isn't met. In that case, there would be a very simple closing, not necessarily in person, when the funds are released to the company from the escrow.
Regulation D for private offerings looks at investments in the issuer over a 12 month period, so often the offering is set up to conclude not less than 12 months after it starts.
But, it wouldn't be at all uncommon for there to be an operating closely held business making a private offering via Regulation D to simply deposit wired funds into its coffers for general purpose liquidity one stock investment at a time, and to continue to do so until 12 months are up (and it possibly starts a new offering), or until has met its goals or can't find anyone else willing to invest.
More generally, real estate closing style closing events are typically characteristic of transactions in which there are more than two parties to the isolated transaction. Real estate closing are as elaborate as they are, primarily, because they usually involve the payoff of the seller's mortgage lender and the creation of a new mortgage owed by the buyer, and because lots of real estate related expenses (e.g. property taxes and utilities) need to be dealt with at the time of the transfer.
But, this isn't typical of a simple sale of initial offering stock to an investor.
In an initial public offering, there is often a formal closing, mostly because the Securities and Exchange Commission and the stock exchange upon which the IPO stock or IPO corporate bonds will be listed have involvement in the transaction in addition to the involvement of the security issuing corporation and the investor in the newly issued securities.