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!