Can I deduct a firearm if purchased through a private party? I am in law enforcement

Yes, you can deduct the cost of the firearm purchased. You should keep documentation of the purchase for your tax records. This purchase would be included in Employment Expenses in Turbo Tax.

Select Federal Taxes
Select Deductions and Credits
Select Show More at Employment Expenses
Select Start or Update at Job Related Expenses

I did receive a copy of dealers record of sale of firearm however there's no amount of how much I paid. Should I request something from the gun shop? And would I deduct the transfer and DOJ fees as well?

Yes, you should have something noting what you paid. You can include the transfer and DOJ fees in the cost of the gun.