I am confused by this field. Should I enter the company that actually issued the dividend or my broker's name?
You should enter the name of the company that issued you the 1099-DIV, so the broker. If you look at 1040 Schedule B, which is why TurboTax is asking, you'll see the instructions along the left say: "If you received a Form 1099-DIV or substitute statement from a brokerage firm, list the firm’s name as the payer and enter the ordinary dividends shown on that form." http://www.irs.gov/pub/irs-pdf/f1040sb.pdf
Note that the payer name should not affect your taxable income or tax due. The IRS primarily cares about the total dollars, which should be reflected on page 1 of Form 1040.