Not being superuser means the export can't be done in a server-side DO block.
It could be done client-side in any programming language that can talk to the database, or assuming a psql-only environment, it's possible to generate a list of \copy statements with an SQL query.
As an example of the latter, assuming the unique output filenames are built from the ID column, something like this should work:
SELECT format('\copy (%s) TO ''file-%s.csv'' CSV', query, id)
FROM table_with_queries;
The result of this query should be put into a file in a format such that it can be directly included into psql, like this:
\pset format unaligned
\pset tuples_only on
-- \g with an argument treats it as an output file.
SELECT format('\copy (%s) TO ''file-%s.csv'' CSV', query, id)
FROM table_with_queries \g /tmp/commands.sql
\i /tmp/commands.sql
As a sidenote, that process cannot be managed with the \gexec meta-command introduced in PG 9.6, because \copy itself is a meta-command. \gexec iterates only on SQL queries, not on meta-commands. Otherwise the whole thing could be done by a single \gexec invocation.
\gexecmetacommand (introduced since 9.6 version)