Hi there! You'll firstly need to ensure your accounts are linked - https://customersupport.frontier.co.uk/hc/en-us/articles/4404175514514-How-to-link-your-Frontier-Account-to-the-platform-on-which-you-play-Elite-Dangerous-
Then, you should just need to use the STEAM LOG IN option on the launcher to access the game. To do this, open up the launcher and select STEAM LOG IN. (You may need to restart the launcher if the button isn't immediately available)
This ensures that the launcher reads your Steam purchases (the standard LOG IN button reads purchases from the Frontier Store only). Doing this should grant you game access :) - Nova