Can I use the annual Alternative Entity Tax for Delaware LLCs as a tax deduction?

Yes, the Alternative Entity Tax is a business expense which can be claimed by the LLC under Taxes and Licenses.

How do I input this in turbo tax? I only see a "estimated tax payments" under "other business expenses".

I am trying to use this deduction for a single member llc with no income.

As you report the income and expenses of the LLC on Schedule C of your personal return, you should enter this tax on Schedule C under Taxes and Licenses. Your Schedule C will show a loss which can partly offset your other income.