According to the manpage of inkscape, it can produce PDF files from the command line:
-A FILENAME, --export-pdf=FILENAME
Export document(s) to PDF format. This format preserves the transparency in the original SVG. Used fonts are subset and
embedded. The default export area is page; you can set it to drawing by --export-area-drawing. You can specify --export-id
to export a single object (all other are hidden); in that case export area is that object's bounding box, but can be set to
page by --export-area-page.
Select the PDF version of the exported PDF file. This option basically exposes the PDF version selector found in the PDF-
export dialog of the GUI. You must provide one of the versions from that combo-box, e.g. "1.4". The default pdf export
version is "1.4".
In combination with a scheduler on your system, you might look every minute for a fresh svg-file, not older than 2 minutes to account for rounding effects, and create the PDF file automatically. Of course it would be reasonable to only look in very limited area for such new svg files.
Another option is inotify, which watches file system events (Linux). I don't know your operating system, but maybe it has something equivalent. This could be used to trigger such a conversion script, too.
If you only need this functionality in some cases, it might be controlled by choosing the directory for the SVG file and restrict the scheduler/inotify to that one.
If that is not an option, it might be controlled by a special startup script, which performs such behaviour and starts inkscape in parallel and performs the pdf-export, sleeping for a certain amount of time between looking for a freshly saved svg.
In all cases it will be useful to know the OS you're using and the question might be concretized on superuser (or AskUbuntu, Unix&Linux, I guess there is a MacOS-section too, somewhere ...).