What you ask for (specifically, 'no keystrokes') is technically impossible.
It is (or, at least, may be) possible to write a script to fill and submit the form, but something has to initiate that script. There is o way that Safari (or Chrome, et. al.) will automatically submit the form without some form of user action.
That said, the standard way of addressing this is actually via JavaScript, not AppleScript. JavaScript is the standard for interacting with web content. AppleScript can function at a higher level (manipulating windows, tabs, menus, etc.), but the nuances of HTML and all that goes into a web page is not its forte.
Now, AppleScript can be used to invoke a JavaScript action, so that's probably your way to go.
Given what you've posted, the following JavaScript should enter and submit the form:
document.getElementById('game-input').value="1234"
document.forms[0].submit()
Which can then be wrapped in an AppleScript do JavaScript command:
tell application "Safari"
tell tab 1 of window 1
do JavaScript "// your JavaScript goes here"
end tell
end tell