Yes, this is possible with some companies.
When you buy shares of stock through a stock broker, the shares are kept in "street name." That means that the shares are registered to the broker, not to you. That makes it easy to sell the stock later. The stock broker keeps track of who actually owns which shares. The system works well, and there are legal protections in place to protect the investors' assets.
You can request that your broker change the stock to your name and request a certificate from the company. However, companies are no longer required to do this, and some won't. Your broker will charge you a fee for this service.
Alternatively, if you really only want one share for decoration, there are companies that specialize in selling shares of stock with certificates. Two of them are giveashare.com and uniquestockgift.com, which offer one real share of stock with a stock certificate in certain popular companies. (Note: I have no experience with either one.) Some companies no longer issue new stock certificates; for those, these services sell you a replica stock certificate along with a real share of electronic stock. (This is now the case for Disney and Apple.)
With your stock certificate, you are an actual official stockholder, entitled to dividends and a vote at the shareholder meeting.
If this is strictly an investment for you, consider the advantages of street name shares:
- With shares in a normal brokerage account, you can sell immediately, if desired. With your stock certificate, you would need to send it back to a broker before selling in the market.
- The broker can hold your dividends in your brokerage account; you don't need to deposit checks from the company.
- The broker handles things like stock splits; with your certificate, if the stock splits, you need to return it and get a new one printed with the correct number of shares listed.
- The broker keeps track of what you own and sends you regular statements to remind you and tell you what your portfolio is worth. You are on your own with the certificate.
- If you change your address, you tell your broker, and you are done. If you hold stock certificates, you need to change your address with each of the companies you hold in order to keep getting your dividend checks.
As to your question on buying stock directly from a company and bypassing a broker altogether, see Can I buy stocks directly from a public company?