The federal courts have ruled that chaplains are a legitimate function of the US government in order to preserve the right of religion, so they are employed in various institutional settings at government expense, to include the military, the Veteran's administration, and in the House/Senate. (Perhaps others...prisons, for example.)
The reason is this: the government has placed people in these institutions, and the nature of the institution itself separate the "members" from regular society in some extraordinary way. Additionally, the controls necessary in the institutional setting require the goverment to have control of persons who regularly interact with the "members."
This is clearly the case with the military, and it is about the military that the federal courts ruled years ago. They decided that the chaplaincy was the "best" solution to the problem of providing the FULL free exercise rights to the members of the military.
As a military chaplain, we regularly spent tax monies for an assortment of soldier support programs.
All legal....and necessary.
I see the same issue in prisons: a) an institutionalized membership, b) extraordianary separation from normal society, c) necessity of institutional control of access to membership.
I'm not sure the military chaplaincy argument holds up when compared to prisons. Prisons and prisoners are far more accessible than are soldiers, particularly if they are on duty in a hostile area.
On the other hand, any citizen can at any time get permission to visit any prisoner with that prisoner's consent (if the prisoner is in good behavior).
In short, the chaplains in the military were a rough solution to very extreme circumstances. However, there do not seem to be such extreme circumstances here, and as others have noted, there are a wide variety of churches who offer such services without using taxpayer money.
Thank you so much for those insights, dear brother in Christ!