Is there a way to automate a button click?

We now have Automations which can run on a schedule (well run in the middle of the night) , but they are limited in the types of steps they can do.  They can edit records, delete records, add records and Run imports.

But Quick Base has big plans for Automations, so I expect to see continuing development in this area.

Even if you ran on a code page and put a schedule in the code you will still need a trigger to activate the code, like pressing the button during the day.  Dan Diebolt would be the person to have an answer to this.

I don't know how complex your script is but if it is simple and only edits, deletes, adds or import records it might be possible to implement the functionality as an Automation. But if the script does more complex processing you need a platform from which that script can run. You can run the script from the browser itself (if you leave the computer on and browser open) using the Crontabs extension

Crontabs Chrome Extension
https://chrome.google.com/webstore/detail/crontabs/abhmbcedbojnghofcfmnageegmkmpkac

or you can employ some type of server solution. 

One attractive solution is to use a FaaS (Function as a Service) service such as Amazon's Lambda, Microsoft's Azure Function, or Google's Function. These FaaS services are considered serverless as all you do is upload the function itself and supply minimal permissions and configuration (you don't have to maintain or configure the server itself). In this case you would select Node as the platform as it can run you existing JavaScript in the FaaS platform.