Where's my invalid character (ORA-00911)

If you use the string literal exactly as you have shown us, the problem is the ; character at the end. You may not include that in the query string in the JDBC calls.

As you are inserting only a single row, a regular INSERT should be just fine even when inserting multiple rows. Using a batched statement is probable more efficient anywy. No need for INSERT ALL. Additionally you don't need the temporary clob and all that. You can simplify your method to something like this (assuming I got the parameters right):

String query1 = "select substr(to_char(max_data),1,4) as year, " + 
  "substr(to_char(max_data),5,6) as month, max_data " +
  "from dss_fin_user.acq_dashboard_src_load_success " + 
  "where source = 'CHQ PeopleSoft FS'";

String query2 = ".....";

String sql = "insert into domo_queries (clob_column) values (?)";
PreparedStatement pstmt = con.prepareStatement(sql);
StringReader reader = new StringReader(query1);
pstmt.setCharacterStream(1, reader, query1.length());
pstmt.addBatch();

reader = new StringReader(query2);
pstmt.setCharacterStream(1, reader, query2.length());
pstmt.addBatch();

pstmt.executeBatch();   
con.commit();

Of the top of my head, can you try to use the 'q' operator for the string literal

something like

insert all
  into domo_queries values (q'[select 
substr(to_char(max_data),1,4) as year,
substr(to_char(max_data),5,6) as month,
max_data
from dss_fin_user.acq_dashboard_src_load_success
where source = 'CHQ PeopleSoft FS']')
select * from dual;

Note that the single quotes of your predicate are not escaped, and the string sits between q'[...]'.