We show that the problem of code construction for multiple access channel resolvability can be reduced to the simpler problem of code construction for source resolvability. Specifically, we propose a multiple access channel resolvability coding scheme that involves randomness recycling, implemented via distributed hashing, and block-Markov encoding, where each encoding block is obtained as a combination of several source resolvability codes. Our construction is independent of the way the source resolvability codes are implemented and yields explicit coding schemes that achieve the multiple access channel resolvability region for an arbitrary discrete memoryless multiple access channel whose input alphabets are binary.