The EVO-STIK League Premier Division side find themselves four points off the play-off places.
In three years under Kittrick, the Seadogs reached three cup finals and won promotion in their first year back in the town.
Chairman, Trevor Bull, said: "Steve has instigated some fantastic times at the football club. I want to thank Steve and Chris for all they have done for Scarborough Athletic.
"Since the end of 2018 our form hasn't been great and have decided that we want to move the club in a different direction next season and we want to start making preparations for 2019/20.
"Each applicant will be judged on their merits, with your elected Board of Directors deciding on who is the best person to take this football club forward into the 2019/20 season".
To apply for the role, please send your Curriculum Vitae (CV) by email in the first instance to: firstname.lastname@example.org