Contrary to the official website and comments in response to the OP, there has been no way to play the current version of RuneScape in the browser for around six months now.
Since 2016, Jagex had been pushing their new native client. Both clients were previously available for download at the same time, but in November of 2019, the links to the Java client were removed and only accessible had you done some digging. A month later, though, in December, the Java-based client was completely discontinued. And, due to protocol changes, previously downloaded versions of the client no longer work either.