We are hoping to find an answer so that no new orders are missed from being dispatched.
Nope, that's a limitation of Neto being web-based. Browser apps don't have the ability to print automatically.
It's not the best way to guarantee orders are being processed anyway - printers are notorious for randomly failing! It's usually better to have a workflow in place to catch missing orders. What status do your orders start in? Do change status after you print them? What about after you dispatch them?
The saved filters in Neto are great - it's easy to create a filter for "all orders in pick which were placed more than 2 days ago". A quick check of that might highlight potential issues.