BBC report that, Ms Rachel Rickert, 27, says she was "shamed" while representing the brand at the New York International Auto Show in April.
Ms Rachel says, she made clear that she needed a toilet break, but was told it was too busy a time. She did not make it in time to change her tampon.
Hyundai Motor America says it is investigating the allegations.
Ms Rickert says she needed to change her underwear and tights, and told her talent rep Erika Seifred what had happened.
She later got a text from Ms Seifred saying the client - Hyundai - wanted her to take the night off.
The model told the news correspondence that she resisted, saying she would rather stay as she was being paid by the hour.
She went to work the next day, 14 April, as normal.
Ms Rickert says the rep then called on 15 April, saying that Hyundai no longer wanted her to work at the show because they heard about her period.
"I was completely surprised," she said.
"I was really upset. I started crying. I book out shows, and I miss other opportunities. So I was just like - 'What? This is not right!'" she said.