Can I use SHAKE-256 on 32-Bit Microcontroller? SHA-512 cannot be really run on a 32-bit ARM microcontroller since it uses the 64-bit constants, but since SHAKE-256 uses only 32-bit values, can it be run on a microcontroller with 32-bit registers?
SHA512 cannot be really run on a 32bit ARM microcontroller since it uses the 64bit constants,

Sure it can; a 32 bit ARM microcontroller can do the 64 bit additions (by performing the addition in 2 32 bit steps and including the carry from one to the other); it can also perform the 64 bit shifts/rotates required by breaking up the operation into multiple steps.
Now, it won't be quite as efficient as it would be if the processor handled 64 bit operations natively, but that's quite different from saying it can't do it at all.

since SHAKE256 uses only 32-bit values, can it be run on a microcontroller with 32bit registers?

Again, the microcontroller is quite capable of performing all the operations required.
What I don't know (and is not immediately obvious) is which would perform faster on an ARM; SHA-512 or SHAKE256 (and that might depend on the size of the input...)