I have an LLC managing real estate property. A non-profit org hired me to manage its property. I hire contractor to work on its property, is 1099 issued by me or by it?

If the NP pays you more than $600 in a year, they issue you a 1099.  Completely separate, if you hire a subcontractor (for any duties) and pay them more than $600 in a year, you issue them a 1099-NEC.

Sure, you can work for free.  You can't take a tax deduction for the value of your time, unless you also report equivalent taxable income.  If you have other business expenses like travel or supplies, you don't take a special charity deduction for that either. You just deduct the expenses in the normal way, and the fact that you don't get paid (and therefore don't have a profit) is where your tax "reduction" comes from.

Thank you very much for the reply and it is very helpful.

I have another question. 

I have this management company.

There are several properties I manage and there are expenses.

I am also looking for new properties to expand the business.

For example, I need to travel to find other properties. 

Where could I enter the expense for this purpose?
In TurboTax, I have to add property first and enter the expense for the property I entered.

Where to enter the expense for looking for new business?

The travel expenses would be reported on your management company's Schedule C of your personal return (or the LLC business return if applicable). Unless you own the rental properties you manage, you wouldn't include Schedule E on your return.