Tobacco giant Reynolds American Inc. submitted an application for some of its Vuse e-cigarettes to the Food and Drug Administration, gaining a head start on other major e-cigarette makers in seeking permission to keep its vaping products on the market.

The Reynolds filing, which totals more than 150,000 pages, gives it a jump on its two main rivals, Juul Labs Inc. and NJOY Holdings Inc. The FDA, which regulates tobacco, has given companies until May 2020 to submit any products they want to keep on store shelves after that...