It's not unreasonable at all.

You can simply mention that you only need the contract signed and the files will be sent as soon as it's done.

The way to ask for this is (reword it better though):

"Files are ready to be sent! Please let me know when you can sign the contract and send it back in order to complete the process, then I'll be able to upload the files to XYZ within the next hour. Also, let me know if there any issue or question regarding the contract and how to add the digital signature on it."

Then you tell them to have a good day, and you sign your name.

And you attach the contract to the email. And you wait!

It's not unreasonable at all.

You can simply mention that you only need the contract signed and the files will be sent as soon as it's done.

The way to ask for this is (reword it better though):

"Files are ready to be sent! Please let me know when you can sign the contract and send it back in order to complete the process, then I'll be able to upload the files to XYZ within the next hour. Also, let me know if there any issue or question regarding the contract and how to add the digital signature on it."

Then you tell them to have a good day, and you sign your name.

And you attach the contract to the email. And you wait!