I opened a PO Box at a pack and mail place years ago to handle mail for a small business that I started. I since closed the business but have several cc's going there. I haven't had any trouble with getting the cards there. I still get junk mail there since the address is on my credit reports.
The one problem I have is that they seem to have about 15 variations of the address on my CRA reports. Fortunately those are easy to correct. Like Lizardking I often just say xx street # XXX and not use the PBM that we are required to use now. The post office made a huge stink about mail not being delivered there without the PBM designation, but the people who own the mail place have no problem with whatever it says. They stick it in my box, without any trouble.
I also get tons of cc pre-offers at the po box, so it doesn't seem to bother them either.