I think all bunny jobs would be volunteer

I do know that some rabbit rescues including some House Rabbit Societies do have a few hired positions, but like Poopy said, most are volunteer.  Have you thought about applying for larger groups, like your local humane society or SPCA.  Though they are animal focused, not bunny focused, many welcome people with knowledge about rabbits!

Also, check out  craigslist.com and in the job section, do a search using "animals, pets, or rabbits"

Good Luck!

Maybe something like that would help you conquer your fears.   I know that a long time ago when I  worked at an animal shelter for a few months, I was afraid of specific dogs, like pit bulls, because of all the news media stuff plus my own personal bad frightening experience ONE time with one. 

But then because I had to work with them everyday, I began to see that the pits that were available for adoption were beautiful wonderful friendly sweethearts.  One was just so friendly that she liked to be held like a baby with her feet up in the air, and she wasn’t small!  Doesn’t mean that I wasn’t normally cautious and looked for signs of aggression, but by exposing myself to them helped me conquer my fear.    At first, I was literally shaking in my shoes, and I was so afraid  and every hair on my head stood up, and sometimes I thought I was going to literally freak.  But I was also afraid of showing that fear in front of the dog for fear that  my own fear would cause them to attack.  So I never acted afraid, and I began to realize I didn’t have any reason to be in the first place.

I was taught how to handle things and it helped me so much, and I found a new appreciation for not only pits, but for big scary type dogs in general.

I remember in the beginning of the interview when I first applied, they asked if there was a particular animal i was afraid of.  I just said, yes…big scary dogs!    They laughed and said, they understood, and that I would be able to have someone help me at first deal with them.   So though I wouldn’t go into an interview saying you were scared of horses, you may rephrase it to say you look forward to gaining more confidence in dealing with horses.  

 I would spend any extra free time I had during the workday picking out dogs I was scared of and walking them – at first I would have someone else come with me, so maybe you can do the same sort of thing – at first just try to hang out with the people that hang with the horses to learn more and feel safe.  

That way you can prevent a great job from slipping away.

Good Luck!