The authorized vaccines do not contain the Covid-19 virus, and you cannot develop Covid-19 from the vaccines. Therefore, the vaccines cannot make you contagious with Covid-19 to anyone you come into contact with.
But it does take time for the vaccines to reach their full protection effects. It's possible to be separately infected with the virus just before or just after vaccination and get sick. Researchers also have not yet determined whether vaccines prevent individual spread of the virus.
That's why wearing masks, social distancing, practicing hand hygiene and avoiding large gatherings remain essential after getting vaccinated.