Can I employ this client on a working app on apex.oracle.com?
I think you can not. Because you probably do have privilege execute dbms_java.grant_permission.
What does prevent implementing such client in pure PL/SQL?
Probably nothing. I think author of package has used Java as it is easiest to implement and provide ready API.
I assume pure PL/SQL do not solve problem what packages you can use in apex.oracle.com.
I'm quite sure that you need e.g. utl_http or similar package that you can not use in apex.oracle.com