Coulson, 32, has been back with the Seadogs since 2017 having started his career with them, playing for the old club between 2004 and 2006.
Kelly told the club’s official website: "For me, Coulo is Mr Scarborough.
"He has been at the club a long time now and he is adored by everyone, from the fans to the players, who all look up to him.
"When I was with him at York City he was always quite reserved, but I have seen another side of him now, how he has grabbed that responsibility by the scruff of the neck.
"He is a fantastic captain and I think he is a fantastic choice for the role."
Coates arrived at the Flamingo Land Stadium last summer, and impressed Kelly when the former Hyde manager took over for the second half of the season.
He added: “I have known Harry a long time and I know the qualities that he has.
"Being a centre-back by trade, I think it is important that you have a central defender as either your captain or vice-captain.
"There are obviously other players that stand out as leaders in the squad, but I'm looking forward to seeing Harry in this role.
"I think he will thrive on the responsibility and I'm delighted to give him the chance."
Meanwhile, Scarborough’s record Football League goalscorer Darren Foreman will lead the club’s physio team next season - joined by Leah Ager and Becky Cox.
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