Will fulfillment and keying the tracking details for the Shopee orders be done on Shopify or Shopee's platform itself?

If you are new to selling on Shopee, most likely you will have to do the fulfillment on Shopee. Shopee has their own system and their own waybills that they want you to use so it's hard to integrate that with Shopify.

Shopee does have a special "Seller's own delivery" option that let's you fulfill with Shopify and we support that. But as far as I know only a few sellers get that from Shopee.