"Do you have any idea..." is more natural than 'an'. Either 'shipped' or 'despatched' would work. As for the choice of modals, it depends how definite the order is. 'Could' sounds quite tentative: you might say this if you haven't placed an order yet. You might say "Have you any idea when my order could be shipped?" if the answer will determine whether or when you place the order: for example, if you order today, it could be shipped this evening. 'Will' sounds more definite. We'd presume that you've already placed the order and you're asking about the schedule. 'Might' is another possibility - less conditional than 'could' but less certain than 'will'.

The more common expression would be the first one. "Do you have an idea when my order could be shipped?" Goods and products are shipped. Generally, messengers and vehicles are dispatched. For example, a taxi or cab company will dispatch cars and send them to addresses to pick up customers. A courier or a delivery person might be dispatched or sent off to make delivery. Hope this helps

Neither is correct. The best way to ask this question would be: Do you have any idea when my order will be shipped?

Do you have any idea when my order will be dispatched Do you have any idea when my order will be shipped Either of these are correct :-)