Is it true that vitamin C can help repair your gums around your teeth?

Yes. Vitamin c is required for the normal health of the gums. Placing it on the gums will have no impact and unless you are low on vitamin c adding extra will have little impact. If you smoke, extra vitamin c can help with the healing but then it would be better to not smoke... If one can stop.
If you have scurvy. Scurvy:deficiency vitamin c deprivation for several weeks to months can cause spongy, swollen gums, tooth loss, bleeding around hair follicles, joint and bone pains, and more. Uncommon since the 1700s. (was treated with ration of lemon juice, or sauerkraut) 10 mg vitamin can treat it. Most people in us are not deficient. Aim for 5+svgs of fruits, veggies/day (try for 9-11).
Scurvy. Once you've developed periodontal disease i'm not certain that vitamin c will repair. However, this though is likely derived from a disease we rarely see anymore that is caused by a lack of vitamin c - scurvy. Among other symptoms, soft, bleeding gums and rotting/missing teeth occur. Interestingly, british navymen were called "limeys" because they were made to eat limes daily to prevent this.

Related Questions

Can vitamin C cause an allergic reaction of mouth? Such as, inflammation of the mouth and/or gums.

Possibly. But it is also can cause sores and inflammed gums if you are taking megadoses (more that is recommended on the bottle). Read more...
Not likely. All food, medications including vitamins can cause allergic reaction but the possibility of allergic reaction to vitamin c is unlikely as it is present in most fruits and vegetables. The lack of vit c causes scurvy with bleeding gums. It is an very old disease seen in sailors who eat no vegetables. Mega dose of vitamins can be toxic to your body! see a periodontist for you gum condition. Read more...