If my veneers were not done properly, meaning there was a slight gap between the veneer and gum would this cause a cavity?

Porcelain veneers without proper fit , can cause gum inflammation, seepage of bacteria, and of course cavities in the tooth and root. The fit of veneers is of utmost importance to bring longevity for the veneers and healthy mouth!

The fit of a veneer is equally as important as the look.  If the porcelain does not fit seamlessly to the tooth, there is always a chance that debris can accumulate potentially causing cavities and gingival inflammation.  The precision of the work and the patient's home care are vital to their longevity.

When the veneers are not well adapted, they will cause inflammation of the gums causing gingivitis and can cause periodontal disease.Cavities will definitely be formed sooner or later, no matter what you do at home: dental floss, brushing, the best solution will be to repair the maladaptation.

Yes, any restoration that is not sealing correctly will very likely give you
problems in the future this problem can be: increase of sensitivity cause by
dentin exposure, accumulation of bacteria for the millimetric gap (not an easy access
to clean well) and therefor possible cavity formation.
Please
make sure you have this problem fixed as soon as possible and, in the meanwhile
be extra careful with your brushing and hygiene technique.

It is unlikely, but yes it is possible depending on how long ago the veneers were done and your home care routine and brushing technique.  If you are not brushing well at the gumline, debris and plaque can remain and cause decay.  Keep in mind however that cavities do not develop overnight, it takes time.  

Yes, if you don’t take care of your teeth daily with rinsing, flossing, and brushing. If you do all that, the uncovered area will be fine. If that gap is visible when you smile, then you need to see if it bothers you esthetically. If it’s a cosmetic issue, then consider to have that veneer replaced. If if that gap is covered by you lips when you smile and you don’t see it, you won’t need to replace it if you take care of your teeth as mentioned above.

If your veneers were not done properly and "there was a slight gap between the veneer and gum" it would cause all kinds of issues because the bacteria that cause all dental disease both decay and periodontal disease have a place to hide and if you can not keep this area perfectly clean you WILL get dental disease.  At our offices in Wilton, CT we believe all dental work should be done so it will last indefinitely and we also teach you how to eliminate the bacteria that cause dental disease.