Air carrier Captain reported that although there were no applicable NOTAMs; the flight crew was informed by ATC that the ILS was unusable because Tower was closed. Due to low visibililty and ILS not usable; flight crew was required to divert.

Narrative

Diverted from intended destination of SRQ due to low visibility. Attempted LOC 14 approach in SRQ but the minimums were too high to break out of the fog. Uneventful diversion with constant coordination between us; Dispatch; and Operations. Report filed.Before departure; I spoke with Dispatch multiple times. During one of our conversations; they mentioned that they called SRQ to inquire about extending Tower hours for our arrival due to an inbound aircraft delay. They denied the request. I asked our new Dispatcher enroute to call SRQ again before they closed for a weather and condition update. They did so and said there were no additional details relative to what we already had. When we checked on with Tampa Approach; they informed us that the ILS was unusable because SRQ Tower was closed. This is the first either us or Dispatch heard of the ILS being unusable when Tower is closed. There were no NOTAMs we could find to indicate this. Both of us were shocked that through two phone calls from two different dispatchers over the course of a few hours; no one mentioned this caveat; especially considering the extremely low visibility and ceilings. Considering that per procedures we are forbidden from shooting the RNAV when the Tower is closed at night; the ILS was our only legitimate likely shot.