Does form 337 need to be sent to the FAA prior or after the work has been performed?
The regulations state that any work must be performed in accordance with either approved, acceptable or specified data on a type certified aircraft. An FAA 337 form is used to report Major Repairs or Alterations to either the Airframe, Engine, Propeller or Appliance. So to answer your question, no, you don't have to send in the 337 form before performing the work because it's purpose is to report the work that was already performed. The form has a conformity statement that must be signed, it states:


  "I certify that the repair and/or alteration made to the unit(s) identified in item 5 above and described on the reverse or attachments hereto
  have been made in accordance with the requirements of Part 43 of the U.S. Federal Aviation Regulations and that the information
  furnished herein is true and correct to the best of my knowledge."


You can view the 337 form here:

https://www.faa.gov/documentLibrary/media/Form/Form_337.pdf