Even though there are means and ways to store data on the browser (not to actual client files) that's not really the general strategy for logging browser errors (which I believe is what you're referring to).
In my experience we'd have a dedicated logging server with a simple API with adequate security that filters traffic and applies rate limiting. That would finally enhance and write logs into a document database that can later be analysed.
A naive JavaScript solution would be using the following to capture errors and send them out to a logging server.
window.onerror = function(message, url, lineNumber) {
// make ajax call to api to save error on server
return true;
};
I should also mention Sentry.io - they provide a service that does this and even though there are some limitations they're usually enough for a small to mid app.
https://github.com/getsentry
