How can I recover my contacts after selling my iPhone?

If your contacts were synced with iCloud you can recover them from there. Log into www.icloud.com and click on the Contacts app there. If there are no contacts there, they were not synced with iCloud (it is an option that you can choose to do or not).

If there are contacts there, see the following from Import, export or print contacts on iCloud.com – Apple Support (UK)

Export a vCard

1. In Contacts on iCloud.com, click to select a contact in the contacts list. If you want to export multiple contacts, press and hold Command (on a Mac computer) or Control (on a Windows computer), then click each contact you want to export.
2. Click  on the right, then choose Export vCard.
3. If you select multiple contacts, Contacts exports a single vCard containing all of them.

Tip: You can export all your contacts as a single vCard and use it as a backup. Click  in the sidebar, choose Select All Contacts, then follow the instructions above to export a vCard.

It's too late to recover your contacts from the phone itself, unless

• You left your personal information on the phone,
• The person to whom you sold it has not wiped that information off yet,
• That person is willing to help you

There is no way for you to remotely retrieve data from the phone itself. If you followed the steps in

What to do before you sell, give away, or trade in your iPhone or iPad - Apple Support

before selling the phone, there won't be any data that the buyer can recover from it, either.

If you were synchronizing your Contacts with iCloud, you could log into the iCloud Web interface using your Apple ID and Apple ID password to see if there is a copy of your Contacts stored on Apple's servers. An iCloud account would normally survive the reset of a device (iPhone, iPad, Mac) connected it.

https://www.icloud.com/

If you were not synchronizing your Contacts with iCloud, but they are in your iPhone's backups in iCloud, I do not know how you would extract the Contacts from one of those backups.