Azure is based on Windows Server 2008 and IIS7. This means that you need to populate the system.webServer part of the web.config file.
The sample file included with elmah's source code contains the details that you need to put in.
<system.webServer>
<validation validateIntegratedModeConfiguration="false"/>
<modules>
<remove name="ScriptModule" />
<add name="ScriptModule" preCondition="managedHandler" type="System.Web.Handlers.ScriptModule, System.Web.Extensions, Version=3.5.0.0, Culture=neutral, PublicKeyToken=31BF3856AD364E35"/>
<add name="ErrorLog" type="Elmah.ErrorLogModule, Elmah"/>
</modules>
<handlers>
<remove name="WebServiceHandlerFactory-Integrated"/>
<remove name="ScriptHandlerFactory" />
<remove name="ScriptHandlerFactoryAppServices" />
<remove name="ScriptResource" />
<add name="ScriptHandlerFactory" verb="*" path="*.asmx" preCondition="integratedMode"
type="System.Web.Script.Services.ScriptHandlerFactory, System.Web.Extensions, Version=3.5.0.0, Culture=neutral, PublicKeyToken=31BF3856AD364E35"/>
<add name="ScriptHandlerFactoryAppServices" verb="*" path="*_AppService.axd" preCondition="integratedMode"
type="System.Web.Script.Services.ScriptHandlerFactory, System.Web.Extensions, Version=3.5.0.0, Culture=neutral, PublicKeyToken=31BF3856AD364E35"/>
<add name="ScriptResource" preCondition="integratedMode" verb="GET,HEAD" path="ScriptResource.axd" type="System.Web.Handlers.ScriptResourceHandler, System.Web.Extensions, Version=3.5.0.0, Culture=neutral, PublicKeyToken=31BF3856AD364E35" />
<add name="elmah" verb="POST,GET,HEAD" path="elmah.axd" type="Elmah.ErrorLogPageFactory, Elmah" />
</handlers>
</system.webServer>
There are 2 elmah lines in the above block of xml that you will need to include, and the section in general should contain most if not all of those elements anyway.
EDIT: No longer required as this is now on by default:
For Elmah to work you will also need to modify the following line in your ServiceDefinition.csdef file:
<WebRole name="WebRole" enableNativeCodeExecution="true">
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