Going through my old answers on Magento StackExchange, I found this question about using the Magento API via JavaScript and noticed that the link to GitHub that contained an important part of the solution, namely implementing a JSON-RPC adapter for the Magento-API is dead now.
So I decided to publish my complete module myself (the original link was a core hack, I rewrote it as a clean module):
The whole module is less than 100 lines of code. In config.xml our controller is added to the api route:
<frontend>
<routers>
<api>
<args>
<modules>
<sgh_jsonrpc before="Mage_Api">SGH_JsonRpc_Api</sgh_jsonrpc>
</modules>
</args>
</api>
</routers>
</frontend>
The new API adapter is defined in api.xml: