One way of achieving this is to create a second website for contact.example.com that contains the contact form on the home page. You can check the Odoo documentation for more details:

https://www.odoo.com/documentation/15.0/applications/websites/website/configuration/multi_website.html

Make sure you setup a reverse proxy (e.g., Nginx or Traefik) to link multiple domain names to your Odoo server. You can check out my blog post on how to configure Traefik and Odoo to effectively set up multiple websites:

https://jortdevreeze.com/blog/odoo-2/a-comprehensive-guide-to-running-multiple-websites-in-odoo-6

I hope this helps!