Karate traditionally has 3 components, Kihon, Kata, and Kumite ... I would try going to some more traditional Karate Dojo and asking if you can practice with them without sparring. Most of the class isn't sparring anyways and if the others spar then you can just practice your Kata or Kihon while they do that. While sparring is important it's possibly the least important component in many traditional schools.

Whether or not you'll be able to progress in belts/ranks is different but who cares. Wearing a white belt forever is probably better for character development. If your goal is the belt just go to the store and buy a belt.

There's also archery and I'm sure there are many other options for no-contact martial arts training. If fencing works for you then there's also Kendo. There's Iai-do which is just about drawing the sword. Lots of options in theory.