Does the app support Oberlo or Dropshipping?

Probably not!

In order to sell on Shopee, you have to use one of their Shipping options. For example:
If you try to ship from China you won't be able to select any of these options. If you cannot ship with one of these options you won't be able move the order from "Ready" to "Shipped" on Shopee.

The last time I checked with Shopee they did not support drop shipping unless you have special permission from Shopee.

The only exceptions are shops that have special permission from Shopee that have the "Seller's Own Delivery" or "Ship from Overseas" shipping options. If you have these special shipping options from Shopee, the app will work with your workflow.