Inside a footer which appears to be constant everywhere on the website, is it best practice to give this option on every single page or should it be ignored on some?
I don't think this is a good idea to repeat this form every time you enter a page, because in my point of view if the user wanted to contact you he knows how to find it, but still, you have the responsibility to show it in a good way.
You can put it:
1- The header but try to make it shine beside the header links on the right side. But it in a creative way (ex: Anything? just Contact us) make the contact us link a link or button whatever you feel good implementation.
2- You can include it as a nice looking button in the footer.
3- Also you can use it as a floating button when you click it, it will open a popup with the contact us fields to fill and send.