Thanks to everyone for good suggestions and input. I was able to cobble this together from a couple of posts. It’s more complicated that I’d hoped but is a lot less than some examples I encountered.
Anyway, it does what I need. Now just need to integrate it into my main script and give it a whirl.
#Load the System.Windows.Forms assembly
Add-Type -AssemblyName System.Windows.Forms
Add-Type -AssemblyName System.Drawing
[System.Windows.Forms.Application]::EnableVisualStyles()
$monitor = [System.Windows.Forms.Screen]::PrimaryScreen
# Define the form object
$Form = New-Object System.Windows.Forms.Form
$Form.Text = "TeamViewer Password"
$Form.Size = New-Object System.Drawing.Size(325,175)
$Form.StartPosition = "Manual"
$form.Left = $monitor.WorkingArea.Width - $form.Width
$form.Top = 10
$form.AutoSize = $true
# Define the input box
$InputBox = New-Object System.Windows.Forms.TextBox
$InputBox.Location = New-Object System.Drawing.Point(50,50)
$InputBox.Size = New-Object System.Drawing.Size(200,20)
$Form.Controls.Add($InputBox)
#Prompt above the input box
$label = New-Object Windows.Forms.Label
$label.Text = "Enter Password from TeamViewer window:"
$label.Location = New-Object Drawing.Point(30, 15)
$form.Controls.Add($label)
# Define the OK button
$Button = New-Object System.Windows.Forms.Button
$Button.Location = New-Object System.Drawing.Point(100,75)
$Button.Size = New-Object System.Drawing.Size(100,30)
$Button.DialogResult = [Windows.Forms.DialogResult]::OK
$Button.Text = "OK"
$Form.Controls.Add($Button)
# Put the box on the screen
$Result = $Form.ShowDialog()
# Check if the OK button was clicked
if ($Result -eq [Windows.Forms.DialogResult]::OK) {
$TVPassword = $InputBox.Text
Write-Host "You entered: $TVPassword"
}
# Dispose of the form
$form.Dispose()
One extra thing for others who, like me, have a limited understanding of all this coding, here’s how to adjust the location.
VERTICAL:
In the Define the Form Object block, change
$form.Top = 10
Higher number moves the input box down from top of the screen
HORIZONTAL:
Same block, in this line:
$form.Left = $monitor.WorkingArea.Width - $form.Width
Change the end to look like this:
$form.Left = $monitor.WorkingArea.Width - ($form.Width + 100)
That will move the window left, away from the right border, by 100 pixels
Oh, and Olaf, it works same on other stuff than TeamViewer. 
(just kidding, I always appreciate your help along my way)
—K