Is there anyway to click the button without user's interaction (by vba or any other way)?
A browser extension could detect you're opening a particular site and then click the button. You could even make the browser extension wait until the textboxes are filled to click it. You can't write a browser extension with VBA, though.
Another possibility that does let you do this with VBA is the user32 library of Windows. You could use the SetCursorPos function to move the mouse to the location of the button and then send a click with the mouse_event function. It depends on the screen resolution and the time it takes the site to load.
There is also the possibility of having a bookmarklet click the button, but that requires the action of clicking the bookmarklet first. Many other instructions could be included there, though, so it would actually be useful if you had to do many other additional things. If you only need to click that button, you could do it directly without the bookmark (unless there is something not letting you click it directly).
If by clicking the image, instead of opening the default browser, you open an IE application, then you totally can click it. It requires the IE instance. It's not the default browser (I guess you're not using IE to browse the internet nowadays).
If by clicking the image, instead of opening the default browser, you open an automatable browser such as Chromium (if using puppeteer) or any of the web drivers of Selenium, then you totally can click it too.
I might be forgetting a few other methods, but basically, manipulating the content of a website requires your to have control of the browser. Just opening the default browser does not give you control on it. If I recall correctly, you can take control of IE even if you did not open it from VBA, but other browsers do not let you unless you open an automatable version of them.
Additionally, you can automate a browser from within using its console. Like, for instance, I'm writing this from Firefox, if I click F12 > console and then I write
alert("hi mom")
and hit enter, I will see the message.