[Someone searches for the client’s keyword, i.e., “hash browns” (not the real keyword), and will be taken directly to the client’s website rather than being given search results. Is this an available method to advertise, how is it done, and, well, how much does it cost?]
Not possible, at least in a legitimate fashion. There have been malware programs that hijack it’s victims’ browsers to automatically redirect to a specified website.
You could have your own malware program authored, either a custom browser, toolbar or browser addon. As long as you fully disclose the malicious behavior in the terms and prompt for agreement, you could get away with it without going to jail, but still, it would be kinda slimy.
The only other way would be to build your own search engine or pay astronomical prices for a position on an existing dubious search engine that basically serves to part you with your money offering little or nothing in return.
Years ago there was just that type of advertisement program available in certain “walled garden” Internet service providers, namely AOL. But all of those types of private networks have faded away as people escaped to the relative freedom of standard Internet access.