Unfortunately, it is not legal to own a monkey as a pet in many areas. Furthermore, wild or exotic animals, such as monkeys, are illegal to own for any reason in the majority of the United States.
However, there are some accredited wildlife rehabilitators and sanctuaries that may be willing to let you visit and/or hold a monkey, typically under close supervision. Checking your local state laws is an important first step. In addition, you can search online for ‘monkey sanctuaries’ and ‘wildlife rehabilitators’ in your area to see if they offer visitation or other ways to interact with a monkey.