Tiktok has been under scrutiny worldwide due to concerns that data could be passed to the Chinese government.
TikTok said it was used by hundreds of thousands of people in Montana.
We want to reassure Montanans that they can continue using TikTok to express themselves, earn a living, and find community as we continue working to defend the rights of our users inside and outside of Montana.
The bill will make it illegal for app stores to offer TikTok, but doesn't stop people who already have TikTok from using it.
TikTok said it has 150 m American users. Although the app's user base has grown in recent years, it is still popular with teenagers and users in their 20s.
In March, a congressional committee grilled TikTok chief executive Shou Zi Chew on whether the Chinese government could access user data or influence what Americans see on the app.
Mr Shou said he never planned to spy on Americans - despite admitting that employees had used the TikTok accounts of journalists to obtain information about them.
The penalties apply to companies, but not individual users. Businesses breaking the law face penalties up to $10,000 8,012, which would be enforced by Montana's Department of Justice.
It means that technology giants such as Apple and Google could be held accountable for fines if they allow TikTok to be downloaded from their app stores in Montana.