Will iPhone 8 earbuds work with iPhone 13?

The iPhone 8 came with Lightning EarPods as well as a Lightning to 3.5mm audio adapter in some countries (but apparently not in the US). I'm guessing this is what the OP is asking about. Apple claims that it's compatible with all Lightning devices (iPhone, iPad, iPod Touch) with at least iOS 10.

EarPods with Lightning Connector - Apple

iPhone 8 - Technical Specifications

In the Box

• • iPhone with iOS 13
  • EarPods with Lightning Connector
  • Lightning to USB Cable
  • USB Power Adapter
  • Documentation

If you have AirPods or AirPods Pro, yes, those will work.

“Earbuds” are what Apple called the old wired earpieces. Those need an adapter, but will work.

The following image shows the AirPods and AirPods Pro earpieces and cases.

AirPods first generation (not shown in the image) and second generation (shown) look identical.

All will work with current iPhone, as well as with other devices that support Bluetooth wireless networking.

Only potential wrinkle with any these is that the batteries may have failed from]\ having been stored. Apple can replace those.

Wired EarPods and other similar wired audio devices will work with a newer iPhone and iPad in conjunction with a Lightning to 3.5mm audio adapter, or with a USB-C to 3.5mm for the various iPad models with USB-C.

Apple removed the 3.5mm audio connector in 2017; with iPhone 7 and later.

Here is the Lightning to 3.5mm adapter: Lightning to 3.5 mm Headphone Jack Adapter - Apple

Pragmatically, that adapter might be worth nearly as much nowadays as are the EarPods that were once included.