How do I authenticate API request when I dont have credentials for the other end

This a very broad question. The right answer for you depends on many factors, and you haven't given us enough information to help you work it out.

Recognise that if you want to service legitimate requests and refuse or deny invalid ones, you are doing authentication, and that means you are evaluating some sort of information about the requester to make your decision. The usual word for that evidence is "credentials". If you are not using a user name and password, you are still evaluating credentials in some form.

Where I would start is to answer the question "what is at stake?". If we are talking about financial information, credit card numbers, sensitive private etc. you need to take more care. If the worst that will happen with an unauthorised request is they discover high scores for some game, maybe you are not so worried about that.

As a couple of people have said in comments, you could filter based on the IP address of the caller.

If the request is arriving via HTTP, there are tools to help. If you are using some other network protocol, you might have to invent similar things for yourself.

You say there is no username or password possible, but I don't understand why that is so. It's quite common to include authentication information in an HTTP header. HTTP itself offers Basic and Digest authentication (https://stackoverflow.com/questions/9534602/what-is-the-difference-between-digest-and-basic-authentication) or you could do your own more customised scheme. If you do settle on HTTP Basic authentication, you really must use Transport Layer Security (TLS), because passwords are sent in the clear so anyone intercepting the request could discover the password.

You could give the caller an API token which they could include in the request, e.g. https://yourapi.example.com/newdataentry?apitoken=12345678 . For example Google Maps does this (https://developers.google.com/maps/documentation/javascript/get-api-key)

You could validate a certificate included in the request. You can generate a self-signed certificate and give it you your caller. You can validate the certificate when you receive it.

If you are willing to host your application on a cloud service, they have tools to help. I am most familiar with Azure but similar services are available on other clouds. Their Identity and Access Management (IAM) services can do a lot of the work for you. For example, on Azure you can create a Security Principal for their application and configure the secret or certificate they are using. You don't need to write the code to authenticate as part of your application. App Services on Azure includes EasyAuth (https://docs.microsoft.com/en-us/azure/app-service/overview-authentication-authorization) which does most of the work for you.