I had this exact problem registering obsolete COM components on Azure. I have registered my methodology here:
AspPDF and AspJPEG for Windows Azure
Igorek is correct, you will need to use the Web role to achieve this. The above solution is based on one web role with running a script to run regsvr32 as a launch task.
, . regsvr32:
chcp 1252>NUL
regsvr32 /s .\library\my-com-class.dll
exit /b 0
ServiceDefinition.csdef:
<Startup>
<Task commandLine="mybatchfile.cmd" executionContext="elevated" taskType="simple" />
</Startup>
.