Be honest and completely communicative. This is in no way unprofessional.
One keynote here is that you're not some random cold caller. You are a user of their website and their service. Or at least you are a user of their website and potentially of their service.
Be open about it. Example text:
Hi, I came across your website looking for local appliance repair—I live in the such and so area. I want you to know that your website's "schedule a visit" feature is broken, and just gives a blank screen. This is more than a little bit frustrating.
I ended up getting my appliance fixed another way, but I wanted to tell you about the broken feature on your website; it might be costing you a lot of business. (I happen to work in web design and may be able to help you with fixing the website, if you don't already have someone to do that.)
Hope you get it fixed okay.
Or, if you are actually a customer of theirs and just impeded by their website rather than totally stopped, something like:
Hi, I've been a customer of yours for a while now but I hadn't tried using your website until today. It's very difficult to navigate. I'm glad I'd already been to your store and love your products; if this website had been my first impression I don't think I would have discovered the quality you provide. (I love my xyz product that I got last year.)
I wanted to bring it to your attention so you can take it up with your IT person and get the website streamlined.
Thanks for the work you do; please keep it up!
(Please let me know if you don't have an IT person currently—I'm not looking for work particularly, but I'm experienced in web design and perhaps we could do business together in other roles. :) Or if not, just know you have a happy customer who thinks your website poorly represents you.)
The keynote is honesty and reality. Don't say something if it isn't true. This is not unprofessional at all; rather the opposite.
If your concept of "professional" is "impersonal; cold; uncaring; unfeeling; robotic phone trees" then I suggest you clear up the word fully. This is the opposite of its true meaning.