A Linux distribution consists of many pieces. All the pieces that are based on software licensed under the GNU GPL and other copyleft licenses must have the code source released. For example, if you ship something built on a Linux kernel, you must provide the Linux kernel source as well as any patch that you have made to the kernel source (however, for the Linux kernel, Linus Torvalds interprets the GPL as not requiring to provide source code for code that is only loaded as a module). You can ship the source code on a CD, or offer that people download it from your website, or any other reasonable method.
You do not have to provide source code for non-GPL programs that are included in the same system. Most distributions (Red Hat, SuSE, Ubuntu, even Debian¹) provide some non-free software in binary form only.
There are other Unix variants that not only do not require open licensing of any core component, but even forbid it. Of course, the flip side is that you'll have to pay to license them. They tend to be operating in the big server realm, not in the embedded realm: Solaris, AIX, HP-UX, SCO... Apple's iOS runs on what is sometimes termed high-end embedded system (MP3 players, mobile phones), but they are exclusively Apple's hardware, you won't be able to license the OS.
There are also unix variants licensed under a BSD license. A BSD license allows you to do pretty much what you want with them, with only a provision that you acknowledge that there is some BSD-licensed software inside (the details of the acknowledgement requirement depend on the version of the license). There are several unix distributions where the whole core system is provided under a BSD license: FreeBSD, OpenBSD, NetBSD are the main ones. Note that some components have different licenses; in particular, the C compiler is GCC, which is under the GNU GPL (you would probably not be shipping the compiler however).
For an embedded system, MINIX is more likely to be appropriate. It is published under a BSD license and designed for both teaching and embedded systems.
A major advantage of Linux is that it has drivers for just about any system you can find. This is not the case with other unices. Even for MINIX, you're likely to have to write a bunch of drivers.
In a commercial embedded system, the value is not in the operating system itself. The value is in integrating all the hardware and software component and making a usable and reliable product out of these disparate pieces. To insist on a free-software-free embedded system, in many cases, is not just reinventing the wheel but reinventing every single part of the vehicle. Concentrate on the part where you're adding value, and reuse what's tried and tested for the rest. Providing source code for GPLv2 components has negligible cost (the situation is a bit more complex for GPLv3, but we're getting widely off-topic).
¹ There is some dispute as to whether the non-free software that the Debian project provides for installation on Debian system are part of the Debian distribution or software that happens to be distributed by the Debian project and packaged for installation on Debian systems. It quacks like a duck, it walks like a duck, and I don't want to get dragged into the dispute as to whether it is a duck.