You can get Element14 to make custom Raspberry Pi's for you:
Unfortunately - For whatever reason (most probably exclusivity agreements with Element14) the 2835/2836 is not available for retail purchase. So purchasing that chip from any other channel except Element14 is impossible for now. These agreements tend to run out after 5 to 10 years and cannot be renegotiated as they then may break unfair trade laws in the EU and US.
If you are making your own circuit based on the Raspberry Pi then yes it is legal. Because the PCB schematics is open source (in a limited form) The problem is... you will not get BCM chips because the manufacturer (Element14) most likely has exclusivity on that chip. I tried and it's impossible (the only way they will be interested is if you buy 500k chips at once.. good luck because Google wanted that in the Chromecast and couldn't even get it)(and don't buy Alibaba based chips because they are fakes or factory back door chips that did not pass QA)
If you are going to be buying Pi boards and modifying them then you still need to credit Raspberry Pi foundation even if you remove the logo's and any traces of anything in any way because those boards have specific licenses on them that are provided by the foundation.
You can use Raspbian without permission from foundation because it is based on Debian. You need to read their license page and credit in the appropriate places. You can use any operating system that runs on ARMv7 as long as you adhere to their license policies. Raspbian just includes some "nice" stuff for education, etc,etc but the packages are available across Linux and BSD now. Plus Windows packages emerging slowly too.
--Edit March 2017
As of this month, the Raspberry Pi is the 3rd best selling computer of all time, selling more than 12.5 million boards in the last five years. That is pretty decent since PC and Mac have been selling computers for 30+ years and take 1st and 2nd place.
I personally don't see Element14 letting go of this gold mine easily.