Is there an effective way to secure access to my API from Salesforce?

I have built an API to my web application so that customers can access certain functionality without going through the dashboard. I have secured it by providing an API key to each customer that identifies them and restricts the IP address from which they can connect. I have now had a request from a customer to allow them to access this API from their Salesforce platform. I don't know much about Salesforce, and when I asked them to let me know their IP address so I could create their API key, all they could give me was the list of reserved IP addresses published by Salesforce. This IP space is huge (millions), covering all of Salesforce, meaning that if someone gained access to the API key they'd be able to use it from any Salesforce account. I have read some things about Salesforce having an OAuth service and having some kind of Application Connect service, but it mostly seems to be designed around allowing 3rd parties to connect to Salesforce - I'm getting a bit bogged down trying to determine if this is any use in my scenario (which requires authentication in the other direction). I'd be grateful for any insight into whether there's something more specific I can do identify a particular Salesforce customer beyond simply putting dozens of CIDR blocks in my API key. I could ask my customer to identify themselves in the referer header when they call my API, for example, but of course that's trivial to spoof. Thanks.