Would a breast lift be possible? I'm 14, and 38DD

Thank you for your question! I hope that these answers help in making
your decision. It is not uncommon to have the symptoms that you
describe, especially in teenagers once her breasts begin to fully
develop. It may be that you have juvenile hypertrophy, in which case,
your symptoms may worsen and breast continue to grow and droop over
time. Once one begins to have the symptoms that you state
above, consideration for a surgical procedure to ameliorate your
symptoms, assist with self esteem, and allow you to get back to physical
activities in your youth should be done. Given your symptoms, you
would be an ideal candidate. You must first discuss with your parents
and pediatrician, and then consulting with a plastic surgeon for
evaluation and examination to assist you in deciding if this would be
the right thing for you. Your surgeon will also go over what to expect
as well as the risks and benefits of the procedure. Overall, your
symptoms should be ameliorated almost immediately and hopefully give you
more self confidence and an increased activity level. Your breasts may
still continue to grow over the next several years, but it is certainly
reasonable to consider this at this age with the issues that you are
having. Many surgeons will recommend awaiting until full breast
development prior to consideration, but discuss your concerns. I hope
that this helps! Good luck with your decision!

at DD cup size a breast reduction may be something for you to consider, although you have to keep in mind that because of your young age your body will continue to grow and change and you will likely need additional reduction/lift sometime in the future.  However, if your breasts are currently affecting your appearance and physical activities, then surgery may be a reasonable option

I'm sorry to hear about the physical and (possible) psychosocial problems your disproportionately large breasts are causing you. 
It is probable that you are dealing with juvenile breast hypertrophy along with the physical and psychosocial consequences of this diagnosis. In my practice, timing of breast reduction surgery is determined on a case-by-case basis. It is not too early to do your homework and learn as much as possible  about  breast reduction/lifting surgery and the potential risks and complications associated with  the procedure.  Hopefully you have the support of your parents as you do your homework.
As you learn about breast reduction surgery, make sure you read about the potential risk/complications associated with the procedure. You may find the attached link helpful in this regard.  For example, unsatisfactory scarring is  one of the potential complications. Make sure you also understand that further surgery may be necessary in the future (for example if the breasts were to grow in size again).
On the other hand, breast reduction surgery is one of the most patient pleasing operations we perform and I think that for the right teenager (enough symptoms) it may be an excellent option (regardless of the age).

Sometimes breast  reduction surgery is covered through health insurance. The best way to obtain insurance coverage for breast reduction surgery involves some “hoops” to jump through. The more documentation you have (for example, from your primary care doctor, physical therapist, chiropractor etc.) the better when it comes to obtaining insurance “authorization” for the procedure.
This documentation and letter/pictures from your plastic surgeon will help you obtain authorization.
 When the time is right,  seek consultation with well-trained/experienced board-certified plastic surgeons.
I hope this, and the attached link, helps.