You can use Post Load user exits to call the rest api and populate the tables accordingly using Business entity calls or SIF calls.
Either we have to go with the BES external call implementation as mentioned in the KB article or we need to go ahead with the PostLoad Userexit.
if you are referring to hosted MDM services, we do allow custom ear deployment there.
Please refer Shriharsha's option which are the only option to achieve it. Your usecase comes under external call purview as it is save button click option.In case you need to the processing post record insert/update then you can use postLoad UE.