Hi Andrew,
As Brooks has highlighted, your client's catering business sounds like a service area business (SAB) and is therefore eligible for a Google My Business listing, provided the staff meets face-to-face with its customers. Your client will need to enter their address when creating the listing and then select the options Brooks as pointed out so that the street address won't be public on the live listing.
That being said, there are some nuances to this. Google did have a bug, in recent memory, during which they briefly showed the addresses of SABs. It was a temporary issue.
Given this, the conversation you need to have with the client surrounds whether their privacy concerns are so extreme that they don't want their street address ever to appear anywhere on the Internet however briefly, or, whether, like most home-based businesses, they just don't want customers showing up at their house thinking it's a walk-in business.
If the former, your hands are tied to doing a limited Local SEO campaign for them and you'll have to rely on Organic SEO, PPC, and Social to get the word out as best you can.
If the latter, then the SAB listing is the solution for this client. It's Google's policy not to show the addresses of these businesses, and you can restrict other citation building to the sites listed in the Phil Rozek article Brooks has linked to. You'll at least be able to do a good, pretty complete campaign for them in this scenario, though not quite as thorough as you can for an SAB that doesn't care whether their address is published on some platforms.
Hope this helps and that you'll have that conversation with the client to discover the source of their objections, whether that be a strong fear or a simple concern over inconvenience.