If you have the cash, you could just buy a business outright https://www.bizbuysell.com/. You could buy the entire business (LLC or Sole Proprietorship) but that's risky since you'd acquire liabilities as well. To avoid that, people often form a new business (Limited Liability Corporation LLC) to buy only the assets of the business. This usually happens for franchises as well eventhough there are extra steps with the Franchisor to qualify and handle the franchise sale.
Specific legal docs will vary by state and maybe as simple as a Bill of sale. Varies other agreements such as leases etc. may also apply. https://www.nerdwallet.com/article/small-business/buying-an-existing-business https://www.entrepreneur.com/article/79638
Larger corporations may buy others outright with cash or stock. Often companies just merge and the larger company will control the merged one and run it.
Being clear on this helps you 1) identify target companies to buy and 2) establish how much you would be willing to pay for the company - and how much it will cost you to extract that value.
Supply and demand will play a large role in how much negotiating power you have. Determining the value of the company is harder when the company is private (there is no efficient capital market that is giving you an instant price, which is available for public companies) but at the end of the day, unless you are buying a public company in a hostile takeover (i.e. you are acquiring a controlling share of this company in the open market without the agreement of its board of directors), the price is what is agreed between the seller and the buyer. You can buy 100% of a company, or a stake in a company. If you buy a majority stake in a company, then you 'own' the business direction because when decisions are put to a vote, your vote will win regardless of what other stakeholders say. However, the return of the business will be shared with the other shareholders. Some small businesses are set up as a sole proprietorship or LLC with only one owner. In this case, you replace the old owner and wholly own the business (or its assets) after the transaction goes through.
There are different ways to buy a company - particularly a small business. You can buy its assets only (then you do not have its liability, such as debt, etc...) or you can buy the entire company (all its assets and all its liabilities). If you are interested in a going concern and a merge, and you want the company to keep operating, the latter might be a good option. Just like title companies for houses, there are companies that help with the paperwork associated with the sale, including commercial banks. It is good to have access to 1) good legal representation, 2) good tax and finance representation.
There are different ways you can pay for a company. You can pay cash, with equity or a mix of both. You can finance your cash in different ways of course such as borrowing it but the seller will get a big check from you. Equity transactions offer happen when a company buys another company. The advantage for the purchased company is that the seller can participate in any upside that might result from the sale of the company (as the value of the stock of the resulting company can go up) as opposed to getting a lump sum for the one time sale transaction. The advantage for the buyer is that it allows non cash rich companies with a high stock value to make acquisitions.
In any case, most decisions on the seller side is a Board-level decision. The Board is the group of people (or the one person!) who is responsible for protecting the interests of shareholders or the current owners of the company. For small companies, the board is simple. If you choose to create an LLC tomorrow, the board will be...you!