Say there is a very rich, very unpopular individual. This individual wishes to buy millions of dollars in stock in a company he or she thinks will be successful. The company, however, realizes that by being associated with this person they will become vulnerable to certain political/social actions such as boycotts.
Can this company block said individual from purchasing the company's stock in the market?
I assume you are talking about a publicly traded company listed on a major stock exchange and the buyer resides in the US. (Private companies and non-US locations can change the rules really a lot.)
The short answer is no, because the company does not own the stock, various investors do. Each investor has to make an individual decision to sell or not sell. But there are complications. If an entity buys more than about 10% of the company they have to file a declaration with the SEC. The limit can be higher if they file an assertion that they are buying it solely for investment and are not seeking control of the company. If they are seeking control of the company then more paperwork must be filed and if they want to buy the whole company they may be required to make a tender offer where they offer to buy any and all shares at a specific price. If the company being bought is a financial institution, then the buyer may have to declare as a bank holding company and more regulations apply. The company can advise shareholders not to take the tender offer, but they cannot forbid it.
So the short answer is, below 10% and for investment purposes only, it is cash and carry: Whoever has the cash gets to carry the stock away. Above that various regulations and declarations apply, but the company still does not have the power prevent the purchase in most circumstances.
A more serious problem: how do you know who's really buying your stock?
"Shell companies" are an increasingly obvious problem in corporate and tax accountability. There are jurisdictions where companies can be created with secret lists of directors and shareholders. If stock is bought by one of these companies, it is very hard to trace it to a particular individual.
In the UK, this is the very definition of a Public Limited Company. A Limited Company can restrict how its stock is trades and who can buy and sell and when, a Public Limited Company cannot.
Most stock exchanges will only allow Public Limited Company stock to be traded.
Therefore a company can control who its stock holders are or be traded on a Stock Exchange.
The company could use registered shares with restricted transferability, i.e. shares that require the consent of the issuing company for a change of ownership.
The answer to this question is given by the fact that many public companies have people who are opposed to the company's aims or practices and who own their stock, often a single share, for the purposes of turning up to shareholder meetings and haranguing directors/asking awkward questions/disrupting proceedings, etc.
If public companies could stop these campaigning shareholders from owning stock they would.
