Other contributors have said "Can you buy stock in trader joes?" is the same question as "Can you buy stock in Trader Joe's's?" If you believe that these are not asking the same thing and should be answered differently, click here
From Trader Joe's directly, no. However, you should check eBay. There are plenty of sellers who peddle Trader Joe's items, and you can email some of them if you can't find what you're looking for...
You're Probably referring to a website that transacts trades for you. Assuming that, I would think that it would trade on the NYSE, NASDAQ, and other U.S. exchanges. That's what American stocks trade...
They usually input the trade on a work station. Sometimes, if the order is large enough (Usually 10000 shares) they will call a trader. A ticket is usually written out after these trades.
The right way to buy a stock is ask the person or company that has the stock in sale to sell it to you then discuss the price and try to get the best price for the stock.