The key is using the Drop app to launch the offer
Not all offers on Drop require you to download the merchant's app, however, if you tap on the Drop offer and are prompted to download/open the merchant app, you can make a qualifying purchase this way!
If you download the merchant's app, please return to the Drop app once it's done downloading. Tap back on the offer and relaunch the merchant app! This will ensure that when you make your purchase, the points post to your Drop account automatically.