SpiderBasic

Is there a way to start a web application in full screen? I know browsers want a button click, but that's exactly what I want to avoid because I want to start in full screen mode right away.

No, you can't. There must always be an user interaction.

Or you can start your Browser in kiosk mode

Dirk stated the only reliable solution. As long as you can expect the user to have a shortcut to the page, you can always start the browser itself in kiosk/fullscreen mode:
C:\Program Files (x86)\Google\Chrome\Application\chrome.exe" --start-fullscreen http://your-spiderbasic-app.com
C:\Program Files (x86)\Microsoft\Edge\Application\msedge.exe" --start-fullscreen http://your-spiderbasic-app.com
C:\Program Files\Mozilla Firefox\firefox.exe" -kiosk http://your-spiderbasic-app.com

Otherwise Peter explained it quite clearly; impossible without user interaction. Or maybe "impossible" is the wrong word, but it's not supposed to be possible. If you happen to find a way (like the old "resize larger than desktop area" tricks etc) it should be considered an "exploit" and therefore extremely unreliable as well as likely soon to be patched. Would otherwise be easy to make webpages looking like a bluescreen, fake desktop windows and so on, and that way manipulate users into different scams.