Yes, the clitoris becomes erect during sexual stimulation. It is much smaller than the penis but the clitoris and the penis have similar nerve endings which give feelings of pleasure when touched during sexual arousal.
The clitoris does in fact get hard when aroused.
It's called vasodilation of the clitoris.
Depending upon your viewpoint, you may find it alarming to know that the female and male sex organs share a number of anatomical similarities. In fact, the clitoris shares a remarkable resemblance to the penis, with similar erectile tissues and a sensitive bundle of nerves in the tip, contained in the glans.
The similarities are more than skin deep!
They are formed during the same phase of development in the human foetus, constructed of the same tissues, and have similar nerve clusters. In the male, the penis testes/sac develop and are pushed from the body to form external adornments, with the penis full of erectile tissue and terminating in the glans. In the female, the same clusters of tissues develop into the ovaries/uterus, vulva/vagina, erectile tissues and clitoris.
They get the better part of the deal, usually, with many times the number of nerves in their clitoris (the female glans is the super-sensitive 'hood' at the top of the inner folds, called labia minora) than the male glans - the most sensitive part of the penis.
Both the glans and the clitoris are projected forward and upward, and made more prominent, by blood-fed erectile tissue. Whereas all the blood goes to the male's penis, the female's blood supply also erects tissues directly behind the vulva, and floods to the tissues surrounding the vagina, whereupon the labia also expand.
The female erection is, in fact, anatomically analogous, from a strict scientific standing, to the male erection, even down to the same tissues being re-tasked to similarly functional structures during early development.