I believe InDesign has a feature to mail merge spreadsheets and PDF forms. There are also paid 3rd party plugins and VBA scripts to do a mail merge from inside Acrobat itself.
Coding a solution is not too difficult - in the past I have used Python to code quick, custom solutions. In the code, you'll need to read each row of the spreadsheet and create an FDF file for the pdf form. Once you have the FDF, you can use PDFTK to merge the original PDF form and the FDF file and create a filled-in version of the PDF form.
If coding isn't your thing, I built a free site where you can mail merge a spreadsheet and a pdf form at pdfzero.com. You can upload a pdf form and a spreadsheet, click on the pdf fields you want to fill in, and select the column you want to use for that field. You can also specify a filename using column data to keep things organized. Once the files are processed, it will return a zip file with all of the merged PDFs.
DISCLAIMER - I wrote Pdfzero