Lightroom does not have a function for saving contact sheets directly to PDF (or any other format), only to print them. So, instead of sending the contact sheet to a printer, it is possible generate a PDF from the print job.

Macs have this functionality built in, but Windows computers will need something like Acrobat installed to be able to generate the PDF.

Alternately on a Mac from the print dialog, instead of saving as a PDF you could also hit Preview and save as JPG or TIFF.

Creating your Contact Sheet in draft mode will drastically improve the speed of creation.