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There are a few ways to do this:

• Inbound Screenpop: This is an approach that will pull up a webpage in another app such as a CRM app or your own app when an inbound call arrives. This is built into some of our out-of-the-box integrations like RingCentral for Salesforce, but you can do it yourself as well. This can be done using the API, using the RingCentral Embeddable UI SDK, and also the RingCentral softphone. We have tutorials for RingCentral Embeddable UI SDK using Salesforce Lightning and performing user searches in Google and LinkedIn. For the RingCentral softphone, you would use the launch external app feature.
• RingCentral Embeddable Integration: The embeddable UI SDK itself can link to external contact databases in which case you could update the inbound caller screen itself.
• Partner Integration: Some of our partners like Tendfold, mentioned by Paul, can assist with these integrations.

Here are some links:

RingCentral Embeddable UI SDK

• https://developer.ringcentral.com/embeddable-voice.html
• https://github.com/ringcentral/ringcentral-embeddable-voice

RingCentral Embeddable UI SDK Tutorials

• https://ringcentral-tutorials.github.io/ringcentral-embeddable-voice-js-quickstart/
• https://github.com/ringcentral-tutorials/ringcentral-embeddable-demos
  

In the last link "ringcentral-embeddable-demos" has demos of Inbound Screenpop in the following demos:

• Static CRM (static CRM)
• Salesforce Lightning (Salesforce popup)
• Salesforce Plus (Salesforce popup plus Google and LinkedIn popups)

Yes, it is possible. But you have to be a little more specific with what "external database" you are thinking of. If, for instance, you'll be using a CRM, then you can definitely integrate it with RingCentral via a CTI. You can check Tenfold, a third-party CTI tool. One of Tenfold's feature is the screen pop, which can show relevant information about a caller when a call comes. It also contains the link to the caller's record in the CRM.

There is API from RingCentral for you to know the incoming call phone number. So that you can lookup the user profile in your external database.