I do not have a delete option when I'm in edit mode with a contact. How do I delete the contact?

If you have scrolled all the way to the bottom and the delete option isn't there, it could be a couple of reasons. Is this a work managed device? If not, it may be a software issue. Have you tried going to iCloud.com? Either way, you may need to restore your device if it's a software issue.

Open Contacts

Select a Contact

Tap Edit in the upper right hand corner

Scroll way way down to the bottom

"Delete Contact" will appear unless other controls have been placed on your phone.

These were already mentioned in the response from ForumTC53.