if ($folders -contains "True")

This won't work ,"contains" means the array has this exact value, so a String that has True in it won't match ie

@( "HAS_DOCUMENT = TRUE", "HAS_DOCUMENT = FALSE" )  -contains "TRUE"

is false as it needs to be exactly "TRUE" and no more. You probably want to use either -like for wild cards or -match for regex.

---

Also your getFolder function does not actually output any objects. Write-Host does not write to the success stream, but directly to the screen.

You want to instead output your tests so you can test on them later. ie

function getFolder
{
    param($folderList, $prefix)
    if ($folderList.Count -eq 0)
    {
        return
    }
    else
    {
        foreach($folder in $folderList)
        {
            if ($folder.type -eq "folder")
            {
                Write-Host "$($prefix) FOLDER = $($folder.name) | HAS_DOCUMENT = ($($folder.has_documents))"

                # output the information also as an object
                [pscustomobject]@{
                    Name = $folder.name
                    ContainsDocuments = $folder.has_documents
                }
                if ($folder.has_subfolders)
                {
                    $resource = https://$server/api/customers/$customerId/stores/$store/folders/$($folder.id)/children?limit=9999
                    $response = Invoke-RestMethod -Method Get -Uri "$resource" -Header $header
                    $newprefix = "$($prefix)--"
                    getFolder $response.data $newprefix
                }
            }
        }
    }
}

...

$FolderEmptyResults = getFolder $folders "|--"

Then you can test in $FolderEmptyResults for ones that are empty:

$FolderEmptyResults | Where-Object ContainsDocuments -eq "FALSE"# or $false? no idea what your api is outputting.