Java7 update 111 and up supports LE. This version is paywalled. Java8u101 and up supports it, and is a relatively painless upgrade, compared to coming from 1.4 or below. You should strongly recommend your client do this, and refuse to support them if they don’t, because Oracle certainly refuses to support those ancient versions of Java.
Java7u21 is 6 years old and full of dozens of easily-exploitable vulnerabilities. You should think twice about even allowing such a client to connect.
That said, you can use keytool to create a custom trust bundle. See https://drissamri.be/blog/2017/02/22/trusting-lets-encrypt-java/ The problem is that while getting one workstation to trust is easy, rolling it out to an organization and keeping it in sync with Java releases is incredibly difficult. You can pass this on to your client, but IMHO they need to upgrade ASAP.