Sit down and talk to them, in person, if possible.
If not, conference call web WebConference, and you take lots of notes. Remember one important fact, don't get overly technical as that's expected of you, never of them.
Start with:
1) Ask for all branding materials (graphics, PDF with logo, marketing literature)
2) Ask for results of any surveys, usability studies or interviews they may have conducted to determine the direction of their web site
3) Ask for a summary of basic expectations (no matter how vague) - never critique it, just accept it and say thank you and keep it as your high level view of what you plan on doing in the future, your "prime commandment". If the client has a detailed listing, the gods are in your favor and use it (see "use cases" below).
You then meet with then again after you do the following:
1) Create a few scenarios describing from a user's perspective what they want to do, and how to accomplish it (referred to as "goal based") and discuss.
Example: You are traveling to Seattle for your job next week and you want to check on the amount you can be reimbursed for meals and other expenses.
It could be a phrase, sentence or even a parapgraph but the key is for you extract your own technical requirements/design considerations based on these short but sweet client scenarios. 10-30 of them is common on a typical job.
2) Discuss each scenario you've created with the client, and create "use cases". Instead of being short and summary, all this means is expand a scenario to include all the details of how users will interact with the site to accomplish that scenario.
Your list of use cases really is the "requirements", i.e. the fundamentals of how the site will function from a user's perspective. Do not discuss design, interface or layout in a use case. Just focus on the sequence of interactions between the user and the web site.
Once you've got that written down, you have your requirements.
Next steps include:
Content inventory
Identification of information auditors (content providers)
Hardware inventory, development platform
Prototyping
and so on...
-jim