He replaces Barrie Hubbard, who has chosen to step down after 29 years on the club's board.
Allen, who had previously held the position of chief executive, told the club website: "I intend to give 100% in my efforts to achieve what the fans want, which is Championship football."
Meanwhile, midfielders Danny Whitaker, 31, and Mark Randall, 18, have agreed new one-year deals.
Whitaker, who joined the Spireites from Oldham in 2010, made 35 appearances in 2011/12.
Randall played 18 games in his first season with the club, having joined them from Arsenal last summer.

Their victims were kidnapped from a bar called Heaven not far from the city's main avenue, Paseo de la Reforma.
Their bodies were found three months later in a mass grave outside the city, after family members organised large street protests.
Prosecutors say they are still questioning dozens of other suspects.
The three men who were sentenced were also fined more than $300,000 (Â£197,500) each.
According to the investigation, the crime was a reprisal for the murder of a drug trafficker which occurred a few days before in another bar in Mexico City.
Seven young men and five women ranging between 16 and 34 years of age were reported missing in the days after the group went to the bar in May 2013.
Surveillance footage showed some of them being led to cars outside the after-hours bar.
There was no obvious sign of force on the surveillance footage. The men who took them away were not masked and did not seem to be carrying weapons.
There was no trace of the missing people until their bodies were discovered in a grave covered with lime, cement and asbestos on the outskirts of Mexico City.
All of them came from the rough Tepito neighbourhood of Mexico City.
The case had horrified the people of Mexico City, which at the time had appeared to be less exposed to the overt drug violence affecting other areas of the country.

It means Blackberry users will have to use the mobile version of the website to log on to the social network.
It is not clear when this will come into effect.
Earlier this month WhatsApp, which is owned by Facebook, also announced that it would only continue to support Blackberry's Android-powered Priv handset.
Whatsapp support for Blackberry's own operating system will end at the end of 2016, the firm said.
"We fought back to work with WhatsApp and Facebook to change their minds, but at this time, their decision stands," wrote Lou Gazzola, who works in developer relations at Blackberry, in a blog post.
"We have worked hard to ensure our end users have the best experience in light of this decision, and are continuing to search for alternate solutions."
Mr Gazzola urged supporters to take to social media using the hashtag #ILoveBB10Apps.
"Hey @facebook and @WhatsApp #ILoveBB10Apps Please don't kill'em *attempts puppy eyes*" tweeted Jo.
"I use BB10 because I wanted an alternative to Android and iOS devices, don't stop offering people an alternative choice! #ILoveBB10Apps" tweeted Ed Hale.
Facebook has been contacted for comment.
Chesterfield have appointed Dave Allen as their new chairman.
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A Mexico City judge has sentenced three men to 520 years in jail each for their roles in the kidnapping and murder of 13 young people two years ago.
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Blackberry said it is "extremely disappointed" with Facebook's decision to discontinue support for its app.