I cannot condone using this as a first resort, but it should work. Generally when dealing with web pages you want to use URL parameters as much as possible since webpage elements are volatile.
With that said you should be able to create an IE comobject and then select the element that way. For example:
First create an IE object:
$ie = New-Object -ComObject "InternetExplorer.Application"
$ie.navigate("YourPageURLHere.com")
$ie.visible = $true
You can manipulate certain elements by ID (ex: name="username") or even type. These can be easily found be viewing the source code of a webpage. Here are a few examples:
$Username = $ie.document.getElementByID("Username")
$Username.value = $UsernameValue
$SearchButton = $ie.Document.getElementsByTagName("button")
$SearchButton.click()
Even if the webpage changes, since we have it all attached to the IE object you can then grab the output elements that generate after the submit button is clicked and feed them back into your script.