It surely won't eliminate any such thing.
However to answer your question, if two folks are residents of the same state they may conduct a sale of a firearm between them without any background checking. (Assuming that their state law does not prohibit that, which few do.) Doesn't matter whether they do the deal over coffee at a diner, in one's driveway, through the local paper classifieds, or over an online service. It is even lawful (in most states) to do the transaction and transfer entirely by mail without ever having met -- again if both are residents of the same state.
This executive action says that IF someone is "in the business of" dealing in guns (which is loosely defined as buying and selling guns as a regular course of BUSINESS, for profit) then using online services to do so does not negate your need to get a Federal Firearms License.
It does NOT somehow restrict private sellers from being able to make deals on line.