It doesn't exist. All nail enhancement products use acrylates. While these do contain the carbon atom and fall under "organic chemistry", they are not organic in the natural sense that you mean.
If you have an allergy, the first step is to visit your GP and get a referral to a dermatologist. You may need to push for this, as some GP are not sympathetic to what they consider to be self inflicted injuries. The dermatologist with test you to find out which ingredients you react to - so take all your product MSDS to guide them. Once you have the ingredient list, you can look for products that don't contain these.
However, if as I suspect, the gel contains Hydroxyethyl Methacrylate (HEMA), then it can be difficult to find a gel that doesn't contain this - especially a soak off gel (not many manufacturers offer these).
When you know which ingredients you react to, you can send me a message here, or visit the following Allergy group on Facebook that I am a member of:
https://www.facebook.com/search/top/?q=nail technicians allergy support group
But the second thing to consider, is how uncured or under-cured nail product is getting on your skin so often that an allergy developed. Because unless you find the solution to this, it is highly likely that you will develop another allergy to any new product - and much faster than before.