The cleanest way is to create 2 subprocesses piped together. You don't need a subprocess for the cat command, just pass an opened file handle:
import subprocess
with open("codegen_query_output.json") as input_stream:
jqp = subprocess.Popen(["jq","-r",'.[0].code'],stdin=input_stream,stdout=subprocess.PIPE)
ep = subprocess.Popen(["echoprint-inverted-query","index.bin"],stdin=jqp.stdout,stdout=subprocess.PIPE)
output = ep.stdout.read()
return_code = ep.wait() or jqp.wait()
The jqp process takes the file contents as input. Its output is passed to ep input.
In the end we read output from ep to get the final result. The return_code is a combination of both return codes. If something goes wrong, it's different from 0 (more detailed return code info would be to test separately of course)
Standard error isn't considered here. It will be displayed to the console, unless stderr=subprocess.STDOUT is set (to merge with piped output)
This method doesn't require a shell or shell=True, it's then more portable and secure.