Try the following code - with some modifications. Essentially idea is to store the CSV file locally and then, upload it to the blob storage(I tested it locally but not from the Runbook, but it should work there as well):
#file name will be a guid string to avoid overlapping
$guid = New-Guid
$guidString =$guid.ToString()
# store csv string to random file with random name (guid)
$LogFull = "$guidString.csv"
$LogItem = New-Item -ItemType File -Name $LogFull
#ignore next two lines as you already have csv string
$Date = Get-Date
$csvstring = ConvertTo-Csv -InputObject $Date -Delimiter ';' -NoTypeInformation
#save csv string locally
$csvstring | Out-File -FilePath $LogFull -Append
#and then upload it to blob storage
#Get key to storage account
$acctKey = (Get-AzureRmStorageAccountKey -Name storage_acc_name -ResourceGroupName EastUS-TestRG).Value[0]
#Map to the reports BLOB context
$storageContext = New-AzureStorageContext -StorageAccountName "StorageAccName" -StorageAccountKey "acc_key"
#Copy the file to the storage account
Set-AzureStorageBlobContent -File $LogFull -Container "your_container" -BlobType "Block" -Context $storageContext -Verbose
Alternative solution with Azure Functions
Try to have your Runbook calling your Http triggered Azure function, and pass the string as the parameter, or in the body. This is just a simple REST API call.
In Azure function, you can have Python, NodeJS or C# code that would put your string to CSV file in the blob store. There are plenty of tutorials for this topic, but first, you need to get your string to AF :)
Have a look at the example below, and try something similar(I have not tested it). Essentially the idea is to invoke simple REST API call and passing your payload in the request body:
[string]$Endpoint= "https://myfunction.azurewebsites.net/api/HttpTrigger?code=my_code_I_get_from_Azure_portal"
. . .
$payload =@" your payload "@
$Header = @{
"Content-Type" = "text/plain";
}
$Result = Invoke-RestMethod -Uri $Endpoint -Method Post -body $Payload Headers $Header
Azure function URL+Code you get from Azure portal, if you click on Azure Function, you will see the button 'Get Function Url'.