Amazon, and most other cookie-based affiliate programs aren’t going to work because most of your traffic is coming from inside the Pinterest app, which doesn’t share cookies with the device’s main browser. That means you won’t get credit for all of the sales that occur during the length of the cookie - for those using the Pinterest browser (80%+ of visitors) you will only get credit for sales that occur right then and there before they make that page go away in the Pinterest browser, which is only a tap away and often happens accidentally.
This problem can also cause problems down the line: let’s say you use Adsense, Advertisers track conversions too, primarily through cookies, and often report conversions back to the Adsense system through Google Analytics to track them. So Google knows what the conversion rate for traffic is from your site, and if you are showing Adsense to in-app Pinterest users, that conversion rate is going to be piss poor because of the cookie problem (even though your traffic may be high quality, those conversions won’t be associated with your traffic), and you may well wind up banned from Adsense. This doesn’t happen to everyone, because it depends on the mix of advertisers they show on your site, what they count as a conversion, and if they use Google’s conversion tracking. But it happens more often than most of us would like to think.
Imo, the best approach if you use Adsense at all is to detect if they are in the Pinterest browser and simply not show it to those users. Requests from the Pinterest browser are easily identifiable, “pinterest” is right there in the agent. It will contain [Pinterest/Android] or [Pinterest/iOS].
As far as other ways to monetize, look at things that are impulse buys, or are an easy conversion that are likely to occur right then and there while they still have the Pinterest browser window open. Mobile app installs work pretty well with Pinterest traffic, because the App Store is one of the few apps that the Pinterest browser will allow to open. Other apps it will open are YouTube and Facebook Messenger (but not Facebook itself) if you use the correct deep linking scheme to open these apps. Anything that is going to take some research on their part before they convert should be avoided.