You are correct; buying a stock with cash is impractical but not illegal or impossible. You cannot transact with an exchange, you cannot transact with a broker

You are correct; buying a stock with cash is impractical but not illegal or impossible. You cannot transact with an exchange, you cannot transact with a broker or a broker/dealer, using cash -- your cash must be deposited somewhere in order for it to be used. You can do a private sale of stock where you give someone cash and they transfer stock that they own to your name, either through a broker or by signing the transfer form on the back of an actual stock certificate. To actually claim ownership of the stock (to receive proxy and dividends) you must transact with a broker dealer or the issuing agent, which will almost certainly mean that you have to deal with AML and KYC regulations.