The first general rule is that a person working out of state (their home state) owes a state tax return to the state where they are working, that only reports income earned in that state. Then, they owe a state tax return for their home state that reports and pays tax on all their world-wide income. If that means they pay tax in two states, the home state gives a credit for taxes paid in the other state.
So the first thing you need to sort out is whether she needs to file an out of state (non-resident) return for the state where she worked. Maybe not, if her income was below their threshold or if she was working in a state with no income tax. If she does have to file an out of state return, you would then need to prepare an amended return for your home state to calculate the taxes owed in the home state, and take advantage of the credit for taxes paid to the other state.
Regarding the penalty, it is another general rule that income taxes are pay-as-you-go, either by withholding or by making estimated payments, and you can be assessed a penalty if you don't have withholding, even if you pay in full when you file. If your daughter owed tax in your home state but did not have withholding in the home state, she was technically required to make estimated payments instead. However, there is usually a minimum amount of tax below which estimates are not required. The letter might have been sent by a computer that did not take into account the small tax bill.
So first, figure out the other state non-resident tax return issue. Does she need to file in the other state? If yes, she needs to prepare an amended return for her home state which may show a reduced tax bill or no tax bill. (In other words, if she owed $9 out of state and $10 in-state on the same internship, the home state would give a $9 credit and she would only owe $1 in-state.)
Then, you can write a letter to the office that sent the notice, pointing out that she only owed $9 and it is under the penalty threshold for your state. (You will have to look that up or tell us what the states are.). You can also ask for a waiver if this is her first year paying taxes or her first time owing a penalty. You can also point that that as she was working out of state, she didn;t realize she would owe tax in two states and didn;t have home state withholding and you will fix that next year.
Now for 2024 planning ahead, she may want to make an estimated payment by September 15 for the taxes from the summer job, that will prevent the problem from coming up and if she over-pays the estimate, she will get the difference back as a tax refund. It will depend on what state she is working in, what her home state is, and whether she is having withholding in one state or both.
