The definition for a bookmaker given by the FreeOnlineDictionary is:
(Group Games / Gambling, except Cards) a person who as an occupation accepts bets, esp on horseraces, and pays out to winning betters
For short, some people refer to a bookmaker as a bookie.
A bookmaker can be illegal or legal, depending on which country the practice takes place in. Wikipedia gives the following information about the legality of bookmakers:
Bookmaking may be legal or illegal, and may be regulated; in the United Kingdom it was at times both regulated and illegal, in that licences were required but no debts arising from gambling could be enforced through the courts. Now, since the inception of the National Lottery, it is not only legal but also a small contributor to the British economy, with a recent explosion of interest with regard to the international gaming sector industry. However, gambling debts were unenforceable under English law until the Gambling Act 2006. Trusted legal bookmakers are members of IBAS, an industry standard organisation to settle disputes.
Bookmaking is generally illegal in the United States, with Nevada being a notable exception. In 2009, one of the co-founders of BetOnline was arrested on bookmaking charges.
In some countries, such as Singapore, Sweden, Canada, Hong Kong and Japan, the only legal bookmaker is owned and operated by the state. In Canada, this is part of the lottery program and is known as Sport Select.
So, when you asked whether there was a specific word for people who take bets on illegal events: No, but a bookmaker's job itself can be illegal. Bookies can help people place bets on any type of event, legal or illegal.
The equivalent of a "broker's firm" for betting would fall under the category of a gambling establishment. Bookies can operate on their own, out of a casino, or in shops. Since there is no one place where you can find them, the group of places you can find them are the gambling establishments.