Yes,
the way to do what you want is to build an External Callout. This is a new(-er) feature available in Apigee X and hybrid. Embed the JDBC logic in the external callout. And connect to that callout with the ExternalCallout policy. This source repo contains an example of a Java server skeleton that implements an external callout.
There is an older feature of Apigee, called "Java callout". This is still supported in Apigee X an in hybrid, but if you want to use JDBC, it's not suitable for your purposes. Using that feature, the Java code runs in-process with the rest of the message processor. That Java callout approach won't work to host your JDBC code. The permissions won't allow it.
With an External Callout, your source code that uses JDBC will run in a separate, external process. In the example given in the repo I cited above, this process is hosted by Google Cloud Run, but you can use anything to host the callout process. Regardless whether you use Cloud Run or some other hosting system, you have complete control over how the process is run, the permissions, and so on. You will be able to connect to JDBC, LDAP, or whatever you like.
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