The act of trapping is not a business or a hobby, it is an service or activity performed by a business or individual.
A business is an entity that is registered with a governmental entity for the purpose of offering products and/or services to the general public or specific individuals/groups. So you can have a business that offers and performs trapping but you cannot have a business that is a hobby.
How the business is registered is what matters in terms of taxes. Sole proprietors must show a profit within a specific amount of time. Failure to do so means that the IRS can remove the business classification and reclassify the revenue as personal versus business income thus increasing your personal taxable amount. To show a profit (income) means there must be money left over after deducting all business expenses (all the money the company pays out to offer and perform services) from revenue (all the money the company takes in from products and services). And don't forget that you still have to pay self employment tax on the revenue so what you see isn't what you get to keep.
Anyone considering starting a business should talk to an accountant to make sure it is worth their time, money, and effort first. Specific to fur tapping, you need to make sure that your fur check not only covers all equipment and licenses but also all your gas, vehicles, and travel expenses. So if you're spending $1,000 in supplies, $1,000 in travel to attend conventions, $3,000 in vehicle expenses/payments and $1,000 in gas after deductions yearly unless your net fur check exceeds $6,000 to cover all of these expenses it probably isn't worth creating a business. However, if you're only spending $2,500 yearly in expenses and your fur check is $6,000 you may want to start a business to take advantage of taxable deductions for vehicles and other high dollar equipment.