Ryder here, from the Social Care Team at Shopify. Welcome to the Community, and thanks for posting! It sounds like you're having trouble with fulfilling your Oberlo orders, and I'd be happy to see if I can help point you in the right direction.
In your Shopify Admin, go to the Apps > Oberlo > Orders page. Here, you'll be able to see your Oberlo-sourced order information and click Order Product to start fulfilling. Only orders that have captured payment will populate here, so, if you haven't already, you may want to set up automatic payment caputres. It's also a good idea to ensure that you've set up your Shopify order fulfillment settings to not automatically fulfill any of the order's line items.
As you can see below, the Order Product button is available when the order's status is To Order.
After placing the order you should see the status change to In Processing and, eventually, to Shipped. You can also track your orders on the Tracking page of the Oberlo side menu, or send tracking information to your customer.
If you've been waiting on any orders for some time, don't hesitate to contact the supplier and follow up on the fulfillment. Just as it's important to provide good customer service, it's important to have open communication with your suppliers to run a successful business. Having trouble finding reliable suppliers? Check out this guide for some tips to consider when searching and working with suppliers that you can depend on.
I hope this helps! Please let me know if you have any questions, or if there's anything I can clarify.