The standard openai package it is possible to add additional fields to the request body. This is particularly useful if you have some kind of proxy which can process additional data for metrics, tracing or FinOps.

Here an example how to do it in the openai package:

import openai
client = openai.OpenAI()

response = client.chat.completions.create(model="gpt-4o", messages = [
    {
        "role": "user",
        "content": "Write a short poem about GitHub"
    }
], 
# it is possible to send additional body fields with "extra_body"
    extra_body={
      "field1": "value 1"
    }
) 

Unfortunately I wasn't able to find a way to mimic this behaviour.

I would suggest to add the feature to provide extra body fields in the RunConfig:
result = Runner.run_sync(agent, input="Say this is a test", run_config=RunConfig(extra_body={"field1":"value 1"}))

Maybe it would also benefit if you can set extra body fields on a per agent basis.
agent = Agent(name="Assistant", instructions="You are a helpful assistant", extra_body={"field1":"value 1"})
These will then be added on top of the extra fields which have been set in the RunConfig.