Is it possible for the gums between my teeth to regrow?

Perhaps: It depends upon why you are missing the gums between your teeth in the first place? I suggest that you see a periodontist to evaluate your specific situation and then you can get an accurate answer to your question. Without know what dental treatment you already received and why you are missing gum tissue, we can only speculate as to what the future holds for you.

Hard to say: If you have recession due to loss of bone and periodontal disease, then the gum tissue will not necessarily regenerate back to pre-periodontitis levels. Consult with a local periodontist who can evaluate your condition.