Quantitative testing on other peoples websites

I want to test other peoples sites, but I want to collect quantitative data, as opposed to qualitative data. For example, I'd like to test the time it takes a user to navigate to a product page once they engage with the navigation. I'd want an average time based on many users.

Is this even possible? Obviously, I would not have the ability to use services such as Optimizely, which would require access to the code.