Find a vector $x$ such that: $\min c^{T}x$ subject to $Ax = b$ and $x\ge0$, with constraints. The definition of a linear program is following:
Find a vector $x$ such that: $\min c^{T}x$, subject to $Ax = b$ and $x\geq0$.
Generally, $b$ is assumed to be a fixed constant. However is it possible to construct a program where values of $b$ are part of the optimization? Could I included $b$ in the optimization by changing $Ax = b$ to $Ax-b = 0$. If so, would I also be able to place constraints upon $b$ like $\sum{b} = 1$ and $1>b>0$? Finally, would such a program be possible to solve efficiently?
I am trying to solve the linear program for Wasserstein Distance between two discrete distributions. In the standard case, b represents the marginals for each datapoint. I know the marginals for the target distribution but the marginals from my source distribution are unknown. I am wondering if there is an efficient way to optimize the marginals for my source distribution such that the Wasserstein distance is minimized.

If you want $b$ to be variable, yes, you can move it to the LHS and change the RHS to $0$.  An optimization modeling language would perform such transformations on your behalf.  If you are instead directly using a solver that requires one constraint matrix and constant vectors for the objective and RHS, you will need to explicitly augment the matrix and vectors to accommodate the new variables.
