Do you need a license to be a veterinary assistant?

That will be different from state to state in the US. In some states veterinary technician is a regulated profession, requiring state level licensing, and in other states it is not. The same will be true for assistants. Contact the professional licensing board of the state of interest to find out more. Even in states where a profession is not regulated, that does not mean that people get into that profession and practice incompetently or unprofessionally. In RI, for example, Veterinary Technician is not regulated. But anyone in RI hiring someone for that work will almost certainly expect high quality hands-on education along with national certification which is independent of state requirements.

So in summary: in states where a profession is regulated, you will need to meet state education and practice requirements in order to apply for and obtain a state license. In states where a profession is not regulated, you may still be asked to meet educational requirements and some certification at the national level, even if they are not required by the state.