@aak1 this is a tricky one but may be doable. Would it be easier to have them put the name of the document or better yet, have the form populate a dropdown of documents available to be updated?

So they would have to upload the document first to the library, then access a new form to set permissions which is how I'm reading it. If this is an option, then you can take the value from the lookup column, find the document ID to that document and run a workflow that sets item permissions to that document based on whatever you needed. 

I don't think that a dropdown will work. Ideally they would be able to link from any of 5 libraries. Each library has thousands of documents. If needed, I could restrict it down to a single library though.

The process is they would upload a document to a library and then go to a list. The list has a workflow that creates a transmittal, sends the transmittal, and then updates permissions on both the list item and the linked document(s). I've basically got everything but the linked document permissions part. I was wondering if I could extract the title of the document somehow, but I'm not sure that is possible at all.

@aak1 I may not be following you somewhat. Care to jump on a quick call to discuss. This should be possible. DM me if you want.