Go High Level API Key

You can, and there's no API key to request from RingCentral for it. Two halves to build:

RingCentral side: register an app in the developer console, use OAuth for each customer account, and subscribe to telephony session or call-log events with a webhook. Your endpoint has to echo back the Validation-Token header when the subscription is created or it will fail silently. Wait for the call-log record to settle (about 40 seconds after hang-up) before you read the final result, or answered calls will show up as missed.

HighLevel side: a marketplace app with a Conversation Provider of type Call. Each call is posted to the contact's conversation with the call endpoint (type Call, status, duration, recording attachment). That call only works with the app's OAuth token, not a private integration token, and the contact must already have a phone number on the record or the write is rejected. One trap: HighLevel's built-in Call Status workflow trigger does not fire for third-party providers, so if your team needs workflows to react to calls, write an outcome tag on the contact and build the workflow on Tag Added instead.

If building it stops making sense, there's a marketplace app that already does this server-side (Ringline, full disclosure, ours).