Anything larger than a freighter is surely a no because full organizations including and especially military ones were destroyed. And I also mean just shils as well. Just about 99% of cruiser and up had to be reconstructed. Initially I immediately said no because it is Imperial. And I would expect someone of the jedi order to have been a Padawan, under a master. A master who was killed, therefore they have no lackeys. 

If you feel your character chose not to go into hiding while crime conttolled the galaxy for a year or two and therefore you gained followers, you must preaent an authentic background. One especially that Sep will agree to.