Are there any hash functions that use integers of arbitrary bit size (e.g., int63)? More precisely, I am looking for hash functions where the internal state can be decomposed as a vector of integers of size 63. That is, machine integers where one of the bits is fixed and unusable.
It would be possible to modify some scheme to move use 63-bit words, but that would require cryptanalysis and new choices for constants. I am not aware of any hash families that would be parametric with regard to word size and allow such uneven values.

Instead I would recommend using any existing hash that internally uses 32-bit words and that does not use many shift or rotation instructions. Modular addition, as well as bitwise operations can be computed on 63-bit values while ignoring all but the 32 low bits.

RIPEMD-160 seems like a decent candidate if 80-bit collision resistance suffices, or perhaps RIPEMD-320 if it does not. Only something like one fifth of the arithmetic instructions are rotations. Even if you need 3-4 instructions to implement rotation, the rest of the function should still dominate runtime.

Note that in the case where the limitation is due to tagged integers losing one bit, it may still be possible to do full 32-bit arithmetic, including rotations, on your 63-bit values without extra performance loss. (Depending on the interpreter/compiler.)