Can I correct the distribution code for my 1099?

Code J is correct.  By itself, having the Roth IRA for 5 years is insufficient.  [Edit] Provided that you meet the IRS definition of "disabled," your distribution is a tax-free qualified distribution.

Reporting this distribution properly using TurboTax requires using forms mode of the CD/download version of TurboTax.  You must manually mark the box labeled "If box 7 code is J or T, check if a qualified distribution" on TurboTax's 1099-R Form for this distribution.  Switch to forms mode, open the 1099-R form, then mark this box.

You can achieve the result in TurboTax.

After entering the Form 1099-R, click the Continue button on the Your 1099-R Entries.  Answer Yes when asked if you opened a Roth IRA before 2013.  That's all you need to do for this income to be treated as nontaxable income. The gross distribution amount will appear on Form 1040 line 15a or Form 1040A line 11a but the entire amount will be excluded from line 15b or 11b.

You will also need to indicate the disability if you are not 59 1/2.

 Please note: A qualified distribution from a Roth IRA is tax-free and penalty-free, provided that the five-year aging requirement has been satisfied and one of the following conditions is met:

• Over age 59½
• Death or disability
• Qualified first-time home purchase

A non-qualified distribution is subject to taxation of earnings and a 10% additional tax unless an exception applies. For Roth IRAs, you can always remove post-tax penalty contributions (also known as "basis") from your Roth IRA without penalty.

When you are entering this information into TurboTax, your Form 1099-R, box 7 codes J, Q and T identifies a Roth IRA distribution and determines the tax treatment. If you have a J or a Q, the distribution is considered taxable unless there is an exception. TurboTax will guide you on all the exceptions

(See the attached screenshot below. Click to enlarge.)