Do high resolution PID chips exist?
If you insist on a single chip solution, then there are a few microcontrollers with 24-bit sigma-delta ADCs on them. Typically 12 bits is enough for the DAC since it's on the other side of the controller (after all, 1 bit can be good enough to get 10-12bit PV control accuracy). Note that single-chip discrete '24-bit' ADCs might give you 18-20 bits ENOB (equivalent number of bits) over a reasonable bandwidth. 24 real bits (0.06ppm) is probably a bit of a pipe dream in most circumstances. 

For example, the ADUCM361. 

If you need the best performance, particularly DC bias and gain stability, you're stuck with separate chips. The speed of micro you need will be determined by how fast you have to do the calculations. At a few Hz, even an 8-bit micro may be fast enough (especially using fixed-point math). If you need to do 1kHz or higher, you might be looking at an ARM or even something with a double-precision FPU on board (single precision isn't quite good enough most likely, at least not without some fiddling). 

It's quite straightforward to implement a textbook PID algorithm but anti-reset windup, gain scheduling, heat/cool, self-tuning and other fixes and augmentations to the textbook algorithm might get a bit more challenging.