There were Reichstag elections in 1933 (twice), 1936 and 1938. There was an election for Berlin's city parliament in March 1933 (as seems to have been the case everywhere in Prussia), but apparently Berlin's municipal parliament stopped convening in June, 1933, and elsewhere these bodies seem to quickly have become meaningless as well. I have not found evidence for local or regional elections after March, 1933.
Does this mean there never were any local or regional elections after 1933? How were mayors and the like appointed in Nazi Germany?
(For some very rough analogies, East Germany was another (essentially) one-party state with no legal opposition and did have local and regional elections roughly every five years. The Soviet Union also seems to have had local elections regularly, though I find it a bit hard to find info on local elections for the Soviet Union as well)