Can I apply for a green card while on an E-2 visa?

The E-2 visa does not necessarily lead to permanent residency or a green card by itself. However, you may find an employer to sponsor you while you’re here. You may fall in love and get married with a US citizen while you’re here or some other opportunity may present itself, for example, also simultaneously qualifying for an EB-5 immigrant investor Visa. It is possible, but it’s just going to be through a different route.