Lookup multiple ids in seperate table with 2 subquerys
You're permitted to do multiple joins in a single statement. You can do this even when joining on the same table by using aliases.

I created a very quick example which is similar to your case. I just used ids instead of a time, and I stored the whole name as a single field; you'll just have to concatenate the first and last name on your own ;)

Schema and Data

CREATE TABLE prtimesheet (
  id numeric,
  submitter_id numeric,
  approver_id numeric
);

 CREATE TABLE srm_resources (
   person_id numeric,
   name text
 );

 INSERT INTO prtimesheet VALUES(1, 1000, 1001), (2,1000,1002), (3,1002, 1001), (4,1001,1002);

 INSERT INTO srm_resources VALUES (1000,'Jack Ryan'), (1001, 'Bart Mancuso'), (1002, 'James Greer');


Multiple joins on same table using aliases

SELECT prtimesheet.id, sub_table.name AS submitter_name, app_table.name AS approver_name
FROM prtimesheet
JOIN srm_resources AS sub_table
ON prtimesheet.submitter_id = sub_table.person_id
JOIN srm_resources AS app_table
ON prtimesheet.approver_id = app_table.person_id
ORDER BY id;


This should fix you right up!

Here is a SQL Fiddle for you to play around with if you like.

Edit: And as suggested...

The Oracle version: Schema and Data

CREATE TABLE prtimesheet (
  id number,
  submitter_id number,
  approver_id numeric
 );

  CREATE TABLE srm_resources (
     person_id number,
     name varchar(25)
  );

  INSERT ALL
  INTO prtimesheet(id, submitter_id, approver_id) VALUES(1, 1000, 1001) 
  INTO prtimesheet(id, submitter_id, approver_id) VALUES(2,1000,1002)
  INTO prtimesheet(id, submitter_id, approver_id) VALUES(3,1002, 1001)
  INTO prtimesheet(id, submitter_id, approver_id) VALUES(4,1001,1002)
  SELECT * FROM DUAL;

 INSERT  ALL
 INTO srm_resources(person_id, name) VALUES(1000,'Jack Ryan') 
 INTO srm_resources(person_id, name) VALUES(1001, 'Bart Mancuso')
 INTO srm_resources(person_id, name) VALUES(1002, 'James Greer')
 SELECT * FROM DUAL;


The Oracle version: Joins and aliases

SELECT prtimesheet.id, sub_table.name AS submitter_name, app_table.name AS approver_name
FROM prtimesheet
JOIN srm_resources sub_table
ON prtimesheet.submitter_id = sub_table.person_id
JOIN srm_resources app_table
ON prtimesheet.approver_id = app_table.person_id
ORDER BY id;


Here is the Oracle SQL Fiddle!