Can I get Veneers if I'm missing teeth?

As long as your bite is stable, veneers at the front should be fine. A photo would help, but your local cosmetic dentist would need to check you occlusion and decide how effectively your canines are supporting your teeth movements and thereby 'protecting' your bite.Veneers are at risk of chipping off it they are not protected when you chew or if you have grinding / bruxing / clenching habit. Even if this is not the case, a combination of veneers and or bridges or implants should get you a great solution. Good luck with getting your smile the way you would like it to be. Kind regardsThomas Dey, BDS, MSc

Veneers need to be bonded to a tooth, so you would not be able to have veneers placed over the gaps where your missing teeth are. You could, however, have veneers placed over any visible teeth that have cosmetic problems such as chips, cracks, or gaps. Although every case is different, lack of back teeth support will add stress to natural front teeth as well as veneered front teeth.
If your main concern is filling the gaps left by your missing teeth, you have a variety of options. Dental implants are the longest lasting and most stable, but they do require surgery. You could also opt for a bridge, which is a false tooth attached to crowns which are placed on neighboring teeth as anchors. 
I recommend visiting a dentist experienced in both general and cosmetic dentistry to discuss your options to get your smile looking great. In the meantime, congratulations on your improved dental hygiene! 

due to the condition you say your mouth presents, provably the best way to go is with crowns and not veneers, that will make them more stronger, and you will have great resistance, instead than with veneers.