Anti-Itch Products
A product with the intended use to treat or relieve itching (an antipruritic) is a drug. This includes itching as a result of bug bites or reactions to poison oak, ivy, or sumac.
If you make any claims that make the consumer think that the intended use of the product is to treat or reduce itching, then the product is a drug.
OTC External Analgesic Products
Anti-itch products are considered external analgesic products. There are non-prescription topical analgesic products to treat itching on the market that often look like cosmetics but are, in fact, over-the-counter drugs. In order to be marketed to treat itching a product must meet the OTC Monograph for external analgesic drug products, using only approved ingredients, label claims, directions, and warning statements.
Reality vs Regulation
There are formulations based on natural substances which have traditionally been used treat itching (e.g. jewelweed, soap and water). However, most of these active ingredients are not on the list of approved ingredients for anti-itch products.
Rightly or wrongly, only products which meet all the requirements in the OTC monograph for external analgesic products may be promoted and sold for the purpose of treating or relieving itching.
Also keep in mind that even if you could get the ingredients and make a product that meets the requirements, in order for the product to be legally marketed it must be manufactured in a facility that is registered as a drug manufacturer with the FDA1 and which follows drug good manufacturing practices as contained in federal regulations.2 It is likely there are state-level drug manufacturing registration and licensing requirements.
1 21 CFR 330.1(b)
2 21 CFR 330.1(a)