This statement has to be true before you can update inventory:
Sum ( Line Items::Reorder Level ) ≥ 1
Strangely, the appearance of the button doesn't change unless this expression evaluates as true:
Sum ( Line Items::Reorder Level ) ≥ 1 and Sum ( Line Items::InventoryPull_Temp ) ≠ Sum ( Line Items::Inventory Pull )
You'll need to examine the script and these fields to see how this controls when you can and cannot update inventory. The most obvious thing to check is to make sure that you have re-order levels specified in the products table.
The inventory tracking in the Invoices Starter Solution isn't the best approach possible. It's vulnerable to user errors creating incorrect inventory levels as a user can create an invoice and then fail to click this update button to take the items listed on the inventory out of inventory. It also appears possible to update inventory, and then change the items ordered in the portal--changes that then cannot be correctly accounted for in this system as you will either be unable to run this script a second time or you may find that you can run this script a second time, but that the update script introduces errors by taking items out of inventory a second time or by failing to put them back into inventory if the item was removed from the invoice or the qty ordered was reduced.
You also don't get any picture of how your inventory levels fluctuate over time which can be very useful when you need to adjust re-order points to reduce the amount of stock tied up in storage without setting levels so low that you risk excessive back-orders/out of stock problems with your customer orders.