I want to make an Apple script to evaluate data

Quite simple - your JavaScript code is never called, so Safari doesn't even bother trying to fill out the form.

This is because you embed the JavaScript code inside a handler:

to inputByName(theName, theValue)

This basically folds everything up to the matching end statement into a single command (in this case, inputByName).

However, to actually run this code you have to call the handler somewhere.

Handlers are most useful for writing blocks of code that are called repeatedly through your script, so you only have to write them once and you invoke them by calling their name. In this case, it's a one-off, so you can either invoke the handler, or (even simpler), just remove the handler definition and embed the code directly in the script

In addition to that, if you want to embed the user_name and user_password variables inside the JavaScript, you need to use concatenation to build the command appropriately. As written the line:

				document.getElementByName('username').value=user_name;

will literally call JavaScript passing 'user_name' as the value (which will likely fail since it doesn't represent a quoted string, nor a JavaScript variable.

Instead use something like:

document.getElementByName('username').value="& theName & ";

where this will combine the JavaScript code with the current value of the AppleScript variable.

Option 1: call the handler:

set user_name to "test_usr"
set user_password to "test_pw"

tell application "Safari"
	activate
	-- this opens the login page
	open location "http://c108-013.cloud.gwdg.de/accounts/login/?next=/sabo/"
	
	-- call the handler to fill in the form
	my inputByName(username, user_password)
	
	-- continue the script flow
	open location "http://c108-013.cloud.gwdg.de/sabo/overview/1"
end tell

to inputByName(theName, theValue)
	tell application "Safari"
		tell tab 1 of window 1
			do JavaScript "
				document.getElementByName('username').value="& theName & ";
				document.getElementByName('password').value=" & theValue & ";
				document.forms[0].submit()
				" in document 1
		end tell
	end tell
end inputByName

Option 2: include the form filling in the normal flow:

set user_name to "test_usr"
set user_password to "test_pw"

tell application "Safari"
	activate
	-- this opens the login page
	open location "http://c108-013.cloud.gwdg.de/accounts/login/?next=/sabo/"
	
	--  fill in the form directly
	tell tab 1 of window 1
		do JavaScript "
			document.getElementByName('username').value=" & user_name & ";
			document.getElementByName('password').value=" & user_password & ";
			document.forms[0].submit()
			" in document 1
	end tell
	
	-- continue the script flow
	open location "http://c108-013.cloud.gwdg.de/sabo/overview/1"
end tell

I guess I needed to dig into JavaScript more. turns out there are two issues there.

The recommended way of setting form values is:

			document.forms[0].elements['element_name'].value='new value';

There are two changes here - one is referencing the form, rather than the entire document. The second is that the value needs to be quoted (in my original example I didn't wrap the value in single quotes). Therefore the JavaScript should look like:

		do JavaScript "
			document.forms[0].elements['username'].value='" & user_name & "';
            document.forms[0].elements['password'].value='" & user_password & "';
	        document.forms[0].submit()"
	end tell

the other thing that you need to check is that you are specifically targeting tab 1 of window 1. This may fail if you have multiple tabs open in this window and it's targeting the first tab while you're looking at the second or third.

This sounds like a simple race condition.

Normally, AppleScripts will execute as quickly as they can. In this case, the script is telling Safari to open a page, and then immediately tries to fill in the form. More than likely the page hasn't finished loading, and therefore the JavaScript fails because it can't find the form elements.

The simple fix? add a delay statement after the open location command to give the page time to load. You'll need to decide how long to wait to allow the page to load.

open location "https://blah.blah.blah/"
delay 5 -- wait 5 seconds to allow the page to load

There's a more complex solution which involves detecting when the page has finished loading, but a simple delay should be all you need.