It's surprisingly easy to be your own privacy service. I run my company's "privacy division" (that's in addition to my role as Chief Sunday Afternoon Movie Outing Strategist) and routinely put the domains we manage behind a privacy wall.
Here's one way to do it. I cleared all this with my domain registrar before doing this (who said that this is fine as long as the contact for the domain can be reached) and you should, too.
1. use your company name for the name of the contact
2. use email aliases or catch-alls for the email contact
3. use a free PTSN-to-VoIP number for the phone contact (I recommend Area775.com)
4. use a mail receiving agent (like Earth Class Mail) for the physical address
This may look like a lot of set up and expense, but everything but Earth Class Mail is free.