We will inform you as soon as we have the documents ready.
(I changed update to inform on the assumption that you would be letting them know that the documents were ready. If you will be giving them more information that you will learn from the documentys, then change it back)
We will update you as soon as we get the documents ready.
The use of "update" means that you have already had conversations in the past with this person, and you are going to re-connect with them as soon as you complete the paperwork. Both parties know the paperwork has to be done before they talk again.