I have red bumps like mosquito bites and they keep appearing. Could it be an allergic reaction?

If it looks like mosquito bites, it could very well be mosquito bites. It could also be a variety of other bug bites, but I can't tell you what because I can't actually see them. They could also potentially be hives, but hives eventually go away after you stop being exposed to the irritant. I would stop using the tanning oil for a couple of days and see if the bumps go away. You should also wear sunscreen!