I couldn't find instructions that didn't focus on using the Contacts app on a Mac or using the iCloud.com interface, so here is one way I was able to export my contacts from my iPhone:
- Open the Contacts app on your iPhone
- Click the "< Lists" button in the top left corner of the screen
- Long-hold your finger on one of the Contact sources/groups on your device (in my case, I had one called "All iPhone")
- The option to "Export" will pop-up
- Once selected, you'll be taken to the standard iOS share screen
- You'll find options to "filter fields" at the top (a way to deselect any specific contact fields)
- From there, you can save or share the generated file ("All iCloud.vcf") containing your contacts by AirDrop, text, or any other apps in your list.
Note: .vcf or vCards are cross-platform and are supported by most address books and e-mail programs.