All internal operations of a perfectly standard-conforming AES implementation can be conveniently implemented over bytes, without carry propagation. There's no security flaw implied. The worst that's bound to happen is that for some operations (like XOR with a 128-bit, 16-byte subkey), things must be repeated over several bytes, and a wider word would reduce the number of individual operations. It is however conceivable that some side-channel attacks (power or emission analysis..) could be more threatening in a bytewide implementation.
Many other modern cryptographic algorithm (SHA-256, SHA-512, Chacha) use rotations and addition over words much wider than 8 bits. They can still be implemented with 8-bit quantities but that's messy (C compilers for 8-bit processors will do this automagically, but often at a large expense in performance for rotation).