It is, of course, impossible to distribute and deliver your photography digitally, unless you use some portion of the digital workflow.
I took the question to mean "Is it possible to be a pro and not use (some) digital today?"
I think that if you are even just maintaining a website with examples of your work, or sending scanned "proofs" to a client, you are using some digital, but you are using it in a way that takes advantage of a facet of digital workflow which is particularly advantageous and generally limited to digital only - it permits the rapid delivery of usable images, as well as sharing those images, almost instantaneously, amongst multiple viewers, in multiple locations.
If the photographic world had evolved in a more rational fashion, analogue and digital might co-exist better. We might be able to comfortably rely on the continuing availability of film and optical printing, because of the quality, and qualities that they offer, while still choosing, if we are so inclined, to provide the additional or new services that digital makes possible.
It is not particularly distressing to me that your average snapshot processor scans and prints digitally - it may actually reduce the amount of handling of negatives, and snapshot processors were often not so great at handling negatives before digital came along. The problem is that the professional market has always relied heavily on the lower prices that were an offshoot of the high volume amateur market, and when that market changed, the professional photofinishing market had to either change with it, or accept markedly higher costs. We now see the results of the choices that were made.