This comes from my Junior Level Linear Systems course and your question reminded me of:
Lathi, Bhagwandas Pannalal. Signal, Systems, and Controls. Intext, 1973.
in his last chapter (7) where he combines discrete and continuous time elements in a hybrid system, the discrete time elements are modeled as continuous time $\delta(t)$ functions followed by a zero order hold. He gives a table (Table 7.4) of "equivalences", and apologizes that $G(s)$ and $G(z)$ is an abuse of notation.
but I don't know how you it's going to help you if all you have is a sequence of numbers, and no functional form, because the Z transform of a finite sequence (necessary to compute) corresponds to a Z transform of a FIR filter that is all zeros.
You could also try fitting an AR or ARMA model to your data and then you have a functional form but it would need to be a very good fit. Any residuals would get you back to the FIR filter Z transform. The Z transform is linear so adding one to another would be OK. The Bilinear transform would get you back to a zero state one sided Laplace.
The 2 approaches FIR and ARMA, will not give the same Z transform and by extension the same Laplace. You need to decide what you want to do with the Laplace and choose accordingly.