Yes you can
Thank you for the post.
"I would like to know if I can upload the database to my server 13 make changes to the data and then hand it back to them to be run on server 12 again."
Evan Rawson is correct (thank you!). Both the FileMaker 12 and FileMaker 13 family use, read, and write to the .fmp12 file format, so as long as only data is being added, then hosting on FileMaker Server 13 to enter data and then moving back to FileMaker Server 12 will work fine.