You can leave $12,500 of your military retired pay out of your Delaware taxable income. Thrift Savings Plan distributions are not part of this exclusion: TSP does not withhold state or local tax, all TSP distributions are subject to Delaware tax, and early distributions may be taxed at a higher rate.
There is nothing to apply for. Claim the $12,500 exclusion on your Delaware personal income tax return using the Delaware Division of Revenue instructions for pension and retirement income.
Your Form 1099-R showing the retired pay, kept with your tax records.