I have succeeded in finding information on a man I believe may be your cousin. I found records for a Lawrence B. Ward on two wartime voyages in two different ships, in 1944 and 1945.
(1) Lawrence B. Ward sailed aboard SS FREMONT OLDER, a U.S. Liberty ship built in 1943, on a voyage from Los Angeles to Prince Rubert, British Columbia, to Seattle, Washington, between April 12 and May 8, 1944.
(2) Lawrence B. Ward sailed aboard SS SAMUEL GOMPERS, another Liberty ship built in 1944, on a voyage from New York to Barry, Wales, and Bordeaux, France, and return to New York, between approximately August 20 and October 21, 1945. (When this voyage began World War II had ended in Europe but the war was still ongoing although about to end in the Pacific.)
I also found Lawrence B. Ward on numerous postwar voyages from 1946 to as late as 1953, mostly aboard SS INDEPENDENCE operating between New York and the Mediterranean, and aboard SS WOLVERINE STATE, between West Coast ports and Japan or Korea. In all cases Lawrence Ward served as a radio officer.
I would not be at all surprised if he had additional seagoing time during World War II or later in addition to the voyages I found. The information I found comes from searches of the subscription website Ancestry.com (www.ancestry.com), which has databases of the names of crew members and passengers aboard merchant ships that arrived in certain U.S. ports of entry following a foreign voyage. Some ports, such as New York, have extensive records at Ancestry.com, while other ports have little or no information at this website. So it is very possible that other voyages in which he sailed may have ended in ports for which Ancestry.com is lacking information.
You may wish to pursue another source of information to find additional details about your cousin’s merchant marine career, by contacting the U.S. Coast Guard. See this web page from the website I manage: http://armed-guard.com/searchmil.html. In particular see section A.2. Records of Individuals – Merchant Marine. You will have to contact the U.S. Coast Guard’s National Maritime Center in Martinsburg, West Virginia. The Coast Guard was and is responsible for issuing certain documents ("seaman's papers") to U.S. merchant mariners, so they should have information about your cousin's time in the merchant marine. You will have to provide as much identifying information as possible about your cousin.
Note that you are not next of kin to your cousin (next of kin = parent, spouse, child, sibling). As such you may not be able to obtain his complete service record. It would be best if someone else in your family who is next of kin could make the request for his service record, for example a child of Lawrence Ward, if such a person exists. You could do the leg work of preparing the request but it would be better if someone who is next of kin to him could actually sign any necessary document(s). If there is no next of kin, make the request yourself and hope for the best. There may be a fee for this service but I expect the Coast Guard would not begin work without informing you of any charges.
Good luck.
Ron Carlson, Webmaster Armed Guard / Merchant Marine website www.armed-guard.com