Indirect contact order

[Hsna]

If the contact order is indirect and limited to email, is it allowed to send a birthday party gift, especially given that no emails have been sent to the children for over four years?

No, I would say not

When was the order made and why have emails not been sent if that was permitted in the order?

I was told that the order is the minimum which is to be provided by the resident parent, not the maximum - ie the resident parent is legally obliged to provide X but it does not mean that the resident parent cannot agree to anything over and above that. I think you would need to consider the best interests of the child.
An order for indirect contact by email is minimal and will have been made for good reasons, I would suggest that the other parent begins by making the effort with that part first, though. How will the other parent know anything about the DC and what their interests are? How will the child know what to make of present from parent at random? On the other hand, maybe it will be important to DC to know that their parent thought of them, even if totally randomly. How old is DC and can they offer an opinion? Not that you need to answer these questions here but that is probably what I would think about.

If there is a court order saying indirect contact only then that is not a 'minimum' that is a maximum and anything outside of that specific approved contact would be a breach. If a court has ordered indirect contact only then there must be a significant risk.

in this case yes, which is why I said it must have been made for good reasons. But my first paragraph is simply what I was told.

A quick Google tells me that parties can agree changes in a child arrangements order but this is informal and if they then disagree, the original order would be binding. So it does come down to what the OP wants to do, and she is the one who knows the background.