9 out of 10 times I just store the data in local storage, but one of my pages uses an embedded form script that allows information(user's name and email) to be passed to it by reading URL parameters but doesn't allow me to pass data to it by altering the form's HTML properties.

Yes this way should work, but unless you use URL Rewriting it's more test.html?email=[DS1=email]

9 out of 10 times I just store the data in local storage, but one of my pages uses an embedded form script that allows information(user's name and email) to be passed to it by reading URL parameters but doesn't allow me to pass data to it by altering the form's HTML properties.

Would I be able to pass dynamic data this way through a setup similar to the image attached?

But I think what you're trying to do is not the best option.
You should store the user email in the local storage, and check it on your page instead. Looking in the URL parameter can became insane to handle from direct links and shared links etc