Is there a connection between sciatica and urinary tract infections?

Not really: Pain due to sciatica may mimic pain due to kidney stone/ colic. Otherwise there is no real relationship.

Not that I know of: Typically not unless there is severe spinal cord injury or chronic opiate use causing urinary retention resulting in chronic infections as a result. This is still very unlikely. Other causes may be iatrogenic, ie chronic steroid injections for sciatica causing immune suppression; however, that is also unlikely.