The shop, on Hamble Lane in Bursledon, was hit twice in a week on May 30 and June 5, Hampshire polcie said.
The man pictured in the CCTV image is seen wearing a black hat worn backwards, a dark jumper and shorts, and white trainers.
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In a statement, a spokesperson for the force said: “We believe the man in this image can help us with our investigation and would urge him, or anyone who knows his identity, to get in touch with us.
“Also, if you witnessed anything of the incidents, saw anything suspicious in the area at the time, or have any CCTV, mobile phone, dash cam or doorbell camera footage which could aid our investigation, please contact us.
“You can do this by calling 101, or by reporting online via the Hampshire and Isle of Wight Constabulary website, quoting incident number 44250258430.”