can you make a fillable form into a custom stamp that can be filled in after being stamped on a document?

No. Once its applied it is flattened and can't be edited as a form anymore.
But you can prompt the user to enter the values required just before the stamp is applied to the document.
Search this site for "dynamic stamps" for more info on this subject.

To make a stamp for use in Acrobat, you start with a pdf image of the stamp.

If you want an interactive or dynamic stamp, you start with the PDF image, add the image as stamp, then add the necessary form fields and scripts. It is even possible to open a dialog box to gather the data for the stamp before placing the stamp. Once a stamp is placed on a PDF that stamp if flattened and the date on the stamp can not be updated or changed.

Creating a Custom Dynamic Stamp by Lori DeFurio
Dynamic Stamp Secrets by Thom Parker

Look at Stamps Gone Wild at PDFScripting.com.