Is it nontrivial to put together a GPS chipset/amp/antenna yourself?
If you mean cloning the actual functionality of the GPS receiver/decoder IC, then no. Not doable in any practical sense. For instance GPS receivers have to correct for Earth's gravitational field by applying Einstein's Theory of General Relativity (no kidding), so most people would find the math to be fairly tough going. It might be possible to get away with just using Euclidean geometry though, if you are not too interested in the absolute positional accuracy of the receiver.
There are also some really tricky timing bits in there as well, not to mention the RF bits wouldn't be easy to fabricate, at least not without a fair amount of experience.
If you have the chipset, then with some PCB fabrication experience you definitely can make the PCB assembly yourself, including the antenna and any controlling MCU you may need. Not completely trivial, but the app notes for the GPS chips frequently explain all the RF details you absolutely need to follow, including the antenna design. So you can often 'steal' the RF design from one chip to use with another, if your chosen GPS receiver chip doesn't provide the needed detail.
Something like this, but with added PCB track antenna, MCU and whatever else you need.