No, I don’t think anyone looking for a stage manager googles “Stage Manager” and hires based on that - but having a put together web site does a bunch of things.
Allows you to put together a more complete sales package other then just a cover letter and resume, and present a strong professional image.
If you own the domain name, allows a more personalized and professional image.
I think, as time goes on, it’s just going to be an expected part of business practice.
I recently DID get a job from me web page. A director was pitching me to a regional theatre I have never worked at, and the googled me, my web site came up, with my resume, production photos - and they were able to get a really strong sense of who I was and the level or work I did - and had the resume in hand IMMEDIATELY to make a decision to “bring me in for an interview”.
Although, funny story, I USED to have a section on my web site that said UPCOMING projects, which listed a show I had booked for the fall, and when they saw that, they initially did not contact me - even though the show they were going offer was a longer contract, 30% more money, possible future life, bigger theatre, bigger director, etc, etc. They only came back to me after some backdoor conversations and when they had an offer pretty much in hand so IF i was interested I could make up my mind and give notice right away.
It worked out VERY well, and I have now deleted that section on my web page.