Does LocalDB support temporary tables?
I can't answer for all versions, but for SQL Server 2012 up until  SQL Server 2017 I am certain that they are supported

LocalDB has the same programmability features as SQL Server Express.


  SQL Server Express LocalDB, a lightweight version of Express that has
  all of its programmability features, yet runs in user mode and has a
  fast, zero-configuration installation and a short list of
  prerequisites.


Source

And then,  building on the previous point, for the SQL Server express 2012 T-SQL Syntax


  Express supports the same T-SQL language elements you find in any
  edition of SQL Server. Not only can you issue data manipulation
  language queries against the database, but you can also run data
  definition language statements to create such objects as views,
  triggers, cursors and stored procedures


Source

Testing (SQL Server 2017)



USE testdb
GO
CREATE TABLE #temp (id int , value nvarchar(255));

INSERT INTO #temp( id ,value)
SELECT 5, 'bla';

SELECT * FROM #temp;


result

id  value
5   bla


The temporary table also works when changing the db's compatibility mode to 100 (2008).