It is well known how could we get a representation of the mapping class group of a surface S (which I assume compact, connected and orientable), given a TQFT. My question is: is there any reference talking about the inverse step: how could we get a TQFT given a representation of the mapping class group of a surface S? How should I think this problem?
The following paper answers your question precisely, I think: arxiv:1408.0668
Specifically Theorem 1.3 (which is elaborated on in Section 4) describes precisely what additional data you must specify to construct a TQFT in addition to representations of the MCG of all genera. Roughly, it is the ability to "glue" together representations of different genera in a coherent manner.