In a word: YES. But the question is really asking about two separate issues. The first question is about what makes a virgin, and what “takes” virginity. The second question is asking about the hymen and how it relates to virginity.
What is a virgin?
This is a complicated question and you might get different answers depending upon who you ask. The technical definition, for a woman, is someone who has not had sexual intercourse where a man's penis penetrates her vagina. So if this is your definition of virginity, then a woman is still a virgin after using a tampon.
What is the hymen? And what does it have to do with virginity?
The hymen is a thin membrane that stretches across the opening of the vagina. This membrane does not usually cover the entire opening of the vagina. The opening in the hymen allows for the passing of menstrual blood, and for the use of tampons.
The hymen has been looked at, in some cultures, as the mark of virginity. In some cases, when a virgin who has an intact hymen has sexual intercourse for the first time, the hymen will tear and bleed. It was thought that if a girl didn't bleed after the first time she had sexual intercourse, she must not have been a virgin. This is completely incorrect. The hymen can stretch to cover most of the opening of the vagina, or it can be very flexible and resists being torn, even after having intercourse. The hymen can be damaged by tampons, by medical vaginal exams, even certain kinds of vigorous physical activity. Whether or not a girl still has an intact (undamaged) hymen, does not indicate if she is a virgin or not. Virginity has to do with sexual activity, not the presence of a hymen!
Behrman, RE, Kliegman, RM, and Jenson, HB. Nelson Textbook of Pediatrics, 2004.