The developer should implement either a Lightning component or a Visualforce page that performs the HTTP REST callout, and use a Lightning Action or a Visualforce quick action to expose the component on the Opportunity detail page. These methods allow the developer to create a user interface that can capture the user input and trigger the callout based on a user-initiated action. The Lightning component or the Visualforce page can use the Apex controller to invoke the HTTP callout using the built-in HTTP classes, such as HttpRequest, HttpResponse, and Http. The Lightning Action or the Visualforce quick action can be added to the page layout of the Opportunity object and configured to display the component in a modal dialog or a panel.
Creating a Process Builder or an after update trigger to perform the HTTP REST callout is not a suitable method for this business requirement, because these methods do not provide a user interface or a user-initiated action. The Process Builder or the trigger would execute the callout whenever the Opportunity is updated, regardless of the user input or intention. This could result in unnecessary or unwanted callouts, as well as hitting the governor limits for callouts.
References: Invoking HTTP Callouts, Configuring an HTTP Callout Action, Create HTTP Callout for Salesforce Flow, Invoking Callouts Using Apex, [Prepare for Your Salesforce Platform Developer I Credential]