Save a PDF with a transparent background?

Only my client on a PC, needs it to be a .PDF with a transparent background

Your client does NOT need a transparent background for print. If you have a big white background that will remain without ink.

If you need to print for example a dark T shirt where you need an aditional white ink as a background of a silouete, send a separate file of that silouete with the instruction that that is the white background.

If you want to open the pdf in a vector based program, the shapes will reman as you have them, so if the file itself does not have a background you can edit it normally.