Your four main options are:
- Flash
- Silverlight
- ActiveX
- Unsecure browser settings (e.g. javascript calling EXEs)
All of the above will be subject to various security limitations or require some acceptance by the end-user. I'm not even certain that the browser calling out to EXEs is even possible any more so it should be avoided at all costs.
Installing an ActiveX control will limit you to IE on Windows, so should probably be avoided. The user would have to agree to installing the ActiveX anyway.
Both Flash and Silverlight are cross-platform (Flash more so than Silverlight) but also require the user to agree to access to the Microphone/Webcam - this is done by the framework, not something you can control/influence. Obviously, if the user does not have Flash or Silverlight installed, you'd need to prompt the user that they need to install it!
As stated by @Oded, there are serious security and privacy issues related to microphone's and webcams, so do not expect future browsers to make this easier for you (e.g. the new family of HTML5 browsers).