The service records of U.S. merchant mariners may be available through the U.S. Coast Guard. While employment records were maintained by the private shipping companies (many no longer in existence) that hired the mariners, as another post stated, the Coast Guard was and is responsible for issuing documents ("seaman's papers") and licenses by which merchant mariners could qualify for employment.
See this page from a website I manage, http://www.armed-guard.com/searchmil.htmlhttp://www.armed-guard.com/searchmil.html, in particular section A.2. Records of Individuals - Merchant Marine. You will have to contact the Coast Guard's National Maritime Center, providing as much identifying information as possible. The information provided by other persons posting responses to this thread should be invaluable to you in attempting to obtain your uncle's records.
Be aware that since you are not next-of-kin to your uncle (next-of-kin being spouse, child or sibling) you may not be able to obtain his record, or at least not the complete record. But give it a try.
As to the census records, other postings in this thread have found him identified as Francis Rynal Nyman. I'm guessing that your uncle was the "Rynal" you mention being in the 1910 census and that he was nicknamed Roy. In the 1910 census he was 7 years old so by the 1920 census he was 17 and may have been on his own.
Ron Carlson, Webmaster Armed Guard / Merchant Marine website www.armed-guard.com