The long requested feature for Weblate is here - it now has API which you can use to control it. It's scope is currently limited, but it will expand in future releases. The API is currently available in Git, deployed on both Demo server and Hosted Weblate, our hosted solution. It will be also part of 2.6 release, which should be released by end of April.
Anyway the API now provides basic operations up to translation level - you can not edit individual translations, handle suggestions or other advanced features. However what you can do includes:
As you can see this really covers the basic operations which were most frequently requested, more will come in future. You can see more detailed the Weblate API documentation.