Hi Eduardo!
Soy de California. I’ve grown up using both words interchangeably. “Vaccine” is something that I’ve heard used more frequently since around 2013 and definitely used more now in reference to the Covid-19 vaccine.
Around my area, it’s common to hear the question “Did you get your flu shot?” when talking about the influenza vaccine.
As explained by other people, “shot” can also mean alcohol, so it’s totally context dependent.
Hope that helps!