Add another third party vendor to Shopify to drop ship

Our nonprofit Shopify store calls a vendor for fulfillment. We want to add another vendor so that they can drop ship the item once our Shopify account notifies them. I will need to know how this will work once the buyer adds one vendor's item to the cart along with another vendor's item to the cart. This needs to be completed as soon as possible. Please tell me how long this will take. Must be fluent in English. Thank you!