jmbarry,
Rinyaujlak R.Cath. records are with Csokonya: https://www.familysearch.org/eng/library/fhlcatalog/supermai...

Civil Registration in Hungary started Oct.1895, still Church Records can be researched after that date in the County Archives, of which Somogy County are most helpful, they actually do research for a very reasonable fee schedule.

Last year did a research in the same general areas, enclosing couple pictures from Tarany.
Hope this helps.
Joseph