Can bacterial vaginitis or yeast infections cause a sense of urgency?

Probably UTI: Very likely you have bladder infection(UTI) otherwise vaginitis by itself should not cause urgency(which is a symptom of bladder infection(also called cystitis). You will need to go see your doctor to find out more and get an antibiotic prescription to control the infection/s).

Sometimes: Sometimes, if there is enough irritation to the anterior vaginal wall, it can give a sense of urinary urgency. However, the most likely source is a developing urinary tract infection or urethritis.