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We introduce and prove convergence of a damped Newton algorithm to approximate solutions of the semidiscrete optimal transport problem with storage fees, corresponding to a problem with hard capacity constraints. This is a variant of the optimal transport problem arising in queue penalization problems and has applications to data clustering. Our result is novel, as it is the first numerical method with proven convergence for this variant problem; additionally, the algorithm applies to the classical semidiscrete optimal transport problem but does not require any connectedness assumptions on the support of the source measure, in contrast with existing results. Furthermore we find some stability results of the associated Laguerre cells. All of our results come with quantitative rates. We also present some numerical examples. [ABSTRACT FROM AUTHOR]