You can insert an SQLExecutor before the FeatureReader to query the database metadata tables. This depends on the database implementation, but for SQL Server it might look like this:
SELECT TABLE_NAME FROM INFORMATION_SCHEMA.TABLES WHERE TABLE_TYPE = 'BASE TABLE' AND TABLE_CATALOG='MySchema' AND TABLE_NAME LIKE '%searchstring%'
You could then pass the resulting attribute TABLE_NAME to the FeatureReader.
Alternatively, you can also use a FeatureReader set only to read the schema features (i.e. no data features), that way you'll get one feature per table / feature class in the source dataset, which you can then filter using a Tester/TestFilter before sending to the second FeatureReader.