Sure, but the most inefficient one. Why buy into all the major drawbacks of a blockchain (essentially lack of speed and abundance of costs) if you can't use any of the advantages (decentralization)?
Private blockchains don't have to have nodes mining 24/7.
They can wait until the mempool is at a certain transaction count then new blocks could be added. Or they could mine 1 transaction blocks as they come in. It doesn't have to work like bitcoin or cryptocurrency in general.
Or maybe its batch-like and mining happens once a day at midnight.
There is no reward for adding a block so there is no race. All miners are known and block minting can be pre-planned.
If you don't mine heavily, the blockchain essentially isn't immutable. The race for blocks is the only reason one can assume immutability. That is the core of the problem: If you don't have a chain that's heavily mined, you don't have the security a blockchain promises. Whats left is an unsecure database that feels like running on a C64.
That is true for Bitcoin and other public blockchains. It is not always true for private chains.
Lets assume 10 suppliers each add 10 new 'produce transactions' on some day. It also occurs during work hours, so maybe between 9-5 there is 100 new transactions.
The miners are all known nodes, in fact, only 10 of them if we assume each supplier gets one.
Once the 100 transactions are minted into new blocks and the day ends...
What exactly are the miners supposed to mine into blocks? There are no transactions. Nothing to add.
Scenario B: Walmart rents a $5 digital ocean VPS and installs postgres. Each producer generates a (e.g. GPG) key pair and sends the public key to Walmart. Each day, each producer creates a new entry in the db, signs it with his/her private key, and is done.
The mining is completely worthless in the scenario you described. Imagine either a rogue producer or Walmart having an interest in changing the database. In your scenario of 10 mining boxes the adversary would just have to buy 5 additional boxes and would have complete control over the blockchain.
