Boxer Hosea Stewart and his partner Aymie Crowther were travelling to her home in Wombourne when they stopped at Morrisons supermarket off the Black Country Route in Bilston last Thursday.
They left their two-year-old Staffordshire Bull Terrier cross French Mastiff, called Rico, in their car as they went inside to buy baby milk at around 12.30pm.
A window was left open for Rico while they were inside and, when they returned, they were horrified to find that he had been taken.
No damage was caused to the car and the pair believe the thief was able to reach inside the window and unlock the door.
'It was hell'
Aymie, aged 23, said: "We never leave him alone as he gets separation anxiety.
"We just went in for some drinks, snacks and baby milk.
"The window was open and we were parked in the shade and in the baby and toddler area, we thought it would be safe.
"When we came back to the car we thought it was unusual that he wasn't jumping up and getting excited.
"When we realised he wasn't there we had a breakdown, it was hell, we didn't know what to do.
"He's never left alone, it was a split second in a shop, and it's our biggest mistake, we never thought this would happen."
Rico recently spent four days in a pet hospital and needs daily medication.
The pair are offering a £1,000 reward for any information that leads to his safe return.
Posts on Facebook have been shared thousands of times, with Black Country residents offering to help search for the two-year-old.
'Please return him home'
Hosea, aged 22, said: "The feeling we had, we didn't know what had happened, it was frustration, it's like losing a family member or child.
"We're extremely worried about him, he needs medication and he's with someone who doesn't know what he likes or needs, and they're obviously not a good person.
"Please, whoever has him, return him home, he needs to be with us.
"We hate to think that he thinks we don't want him, he hasn't seen us in days, it's heartbreaking.
"We're overwhelmed with the support people have given us in sharing posts and searching for him, we're being flooded with messages.
"Please just keep looking and help get him home."
Aymie added: "We haven't stopped walking the streets and searching for him, asking everyone if they've seen him and looking for information, we're desperate.
"Thousands of people are sharing it on social media, we hope it will help find him, we will not stop until we find him.
"It makes me sick to the stomach knowing that someone could do this."
Anyone who spots Rico or has information on his whereabouts can contact Hosea on 07577066161 or Aymie on 07712341249.
West Midlands Police has confirmed they are investigating the incident and are making enquiries.
They asked anyone with information to contact them quoting crime reference number 20wv/152052a/19.
Morrisons has been contacted for comment.