Editing my reply:
First, you must list your charitable contributions in the federal section of Turbotax, even if you don't itemize, in order for the deduction to flow downhill to the state return.
However, this creates a situation where you make a donation in 2019, and get it fully repaid in 2020. This may be taxable on your federal return, although I haven't seen this specifically addressed before. That's what the rest of my answer below describes.
That depends on if you are talking about a deduction or an actual dollar for dollar credit.
If you listed a charitable donation of $100 on your 2019 return, and your state actually pays you the full $100 back as a credit, I would say that is a "taxable recovery", or in other words a reimbursement of a previous deduction, and it would be taxable on your return for the year in which the payment was made (2020 in this example).
Recoveries are not always taxable, you must follow the tax benefit rule. For example, suppose you list $1000 of deductions on your federal return, but your itemized deductions are less than the standard deduction. Then the $1000 of deductions flows to your state return and you get a direct $400 credit. That $400 is not a taxable recovery on your federal return because you did not get a tax benefit -- your standard deduction would have been the same if you had claimed $1000 of donations or $600 of donations. On the other hand, suppose you had $1000 of charitable contributions and your total itemized deductions were $16,000 including taxes and mortgage (and you are single). Then, the tax benefit rule applies, because you got a better deduction ($16,000) than you should have ($15,600), so the credit would be treated as federal taxable income.
This is not reviewed in Turbotax, probably because it is a rare situation; only a couple of states give direct credits for contributions instead of a deduction. You can enter taxable recoveries in the "other income" section at the bottom of the income page.
However, this added income will flow back to your state return, but it should not, because it's not taxable 2020 income in your state. This would require making a manual adjustment on your state return using Turbotax installed on your own computer--it can't be done online, and most manual adjustments will void the accuracy guarantee.
So I think that, while technically this could be a taxable event, I suspect you would be in large company if you didn't mention it, and the IRS has too many other problems to worry about auditing you over $400.
