Pending this Pull Request https://github.com/composer/composer/pull/1341 you can make Composer run on Azure as post deployment command.
The way Azure Websites works, this doesn't affect your site at all. The composer installation is run, and after that completed successfully, the webroot is symlinked to the new checkout.
To get this working before the PR is merged, checkout my composer fork, call
./bin/compile and use the generated composer.phar file as part of your git repository.