I want to have page in across all SharePoint sites in Office 365. I know I can't use pages on the
_layouts, but my goal is to create page for users to see and get support information to contact the organization. An application page will be great to achieve the result.
One option would be to go into your Office 365 Admin portal and go to Company Profile. There click on the Company Help Desk link. Here you can add some help information like phone and email, and provide a URL. This can point to anywhere, on prem or a single page you host somewhere else in your tenant that everyone has access to, a knowledge base, or your own help desk software.
Then this information is in the ? menu of the suite bar across your entire Office 365 workloads.