How do you get Automator to click a button on a web page?

Automator does not provide robust actions for this sort of task. If you learn enough AppleScript, you can manage it, but AppleScript takes time to master.

Or you can get the handy Fake, http://fakeapp.com/ --a good investment if you think you'll be doing this kind of thing often.

I know what you need to do. Go into Automator, and then click the record button on the toolbar (see screenshot below). After that, you should see a black window that says 'Automator Recording' and has a stop button in it. Now go into Safari and click away. After you are done, click on the stop button in the black window. Now you can save your workflow and click the run button.

Watch Me Do may work, but it's also likely to fail. The action records mouse and keyboard actions as they take place on screen. If the (aptly abbreviated) WMD action doesn't find the expected screen elements--everything just where it first was when the recording was made--it will usually bomb. In other words, WMD doesn't, for instance, find fields or buttons on a webpage, it simply navigates to a screen location and tries to type/click as it has been instructed in that spot. A few pixels off, boom!

As for AppleScript, here's one place to begin: http://www.macosxautomation.com/applescript/learn.html. Look at the Beginner's Tutorial "start here" link materials.

Explore the Mac OS X Technologies subforum, https://discussions.apple.com/community/mac_os/mac_os_x_technologies. It's where the AppleScript mavens can be found. If you search that forum for "learn AppleScript," you'll find great advice.