Advise from my experiences
+1 on getting a pellet gun/air rifle
- Get a bigger dog. Small dogs/cats are as much prey as predator for foxes. If you don't want a permanent second dog, consider fostering a big dog through a rescue organization. Hint (a breed such as Fox hound or Fox Terrier may be a good place to start).
- Remove their food sources: garbage, rabbits, field mice, etc. Having a hawk or an owl in the neighborhood provides good competition for the fox population. Foxes are smart. Around here they know which nights people put out their trash.
- Get a trail camera (or two or three). What you want is a trail camera with a bright motion activated flash. This and motion detecting lights in general will help.
- Unfortunately, I have had no success with humane traps (only kind allowed near me). The Habi-trap that I bought to catch the invading raccoon's and foxes, only caught one squirrel. The problem was in releasing a very upset squirrel without getting myself bitten. Local animal control is not that much help. If you do go the trap route, a trail camera will help you get the correct size.
- You need to get your neighbors on your side. Putting out food encourages the full spectrum of critters and just creates a dependency on humans.
- In the UK, there apparently is a night hunting technique called "lamping". You might be able to entice a local hunter to try a night hunt on your property or ideally with your neighbors cooperation throughout the neighborhood. Probably a good idea that your children have a sleep over with friends that evening.
- A big fence
- Roll with it and be grateful that you have your rodent and rabbit population under control. Also that you don't have a wolf or bear problem.