You'd need to design, or have designed, an application-specific motherboard for it -- the kinds of higher-end SOCs you find in phones and tablets, like Tegra chips, are sufficiently complex that a hobbyist has no hope of designing such a board. If you're not doing it professionally already, DIY is not an option.
However, if you had the motherboard sorted, you'd source your parts from somewhere like Digikey (for prototyping / low-count production) or direct from the manufacturer for high-volume products. Sometimes SOC designers offer samples for prototyping purposes, but you usually have to be an established relationship with the capacity to deliver on your designs.
For chips that are less complex -- simpler ARM-based microcontrollers -- those kinds of designs are within the reach of a dedicated hobbyist to design and have produced, but those will generally be far less capable devices.
If you look around, you can find pre-built evaluation and prototyping boards that offer chips of various capability, from very small to very large (comparable to Tegra3/4, possibly even Tegra4-based boards). You could in theory develop a design around one of these boards, the trick is finding one that offers the interfaces you need, and then developing the software and outside hardware interfaces necessary. If the protype was successful, you can order a batch of boards for limited-production, or for high-volume production, work with someone to adapt the boards reference design to your needs and reduce cost by eliminating unused features.
Another option would be to base the machine on a low-cost commodity platform like an ITX PC motheboard and an inexpensive, low-power x86 processor, either with integrated or discrete graphics. There are likewise commercially-available peripherals for interfacing standard JAMMA arcade cabinets from commodity PCs. PC processors with integrated graphics get looked down on, and aren't as sexy as saying "Tegra4 inside", but even Intel's latest integrated GPU is far more capable than Tegra4. AMD's brand-new-as-of-today Kabini APUs are even more capable.