I doubt this shooting is going to change anything. It's always the same spiel, isn't it?
Mass shooting -> HUGE outrage -> intense discussions on gun laws -> discussions fade into the background after 1-2 months.
Nothing is going to change. Especially not under Trump's administration.
Nothing would change for years even if federal gun laws were drastically changed overnight. There are more guns than people in the US, even if gun and ammo sales immediately came to a halt right now, there is still enough of a stockpile for years to come.
I can imagine there will be legislation regarding bump fire stocks, like Slide-Fire, which can basically reach the same rate of fire as a full-auto without modifying internal components, thus circumventing the legal process of obtaining a full-auto weapon (Class 3). It's believable that these types of stocks will eventually be regulated the same as other class 3 items like full-auto, short barreled rifles/shotguns, suppresses and destructive devices (>.50).
Most legislation proposed after these incidents focuses solely on cosmetic and mechanic features of the weapon, unfortunately... rather than requirements of the person trying to obtain the weapon. I've seen many, many people who go buy a gun who simply shouldn't be around them. There is a staggering amount of irresponsibility, stupidity and even bad intentions.
Background checks are required at a federal level for anyone buying from a seller with a FFL, this will be the same buying from a shop in any state.. same form, same call to the FBI (with the exception of ccw holders in some states). But, some states allow private sales between citizens, which is tricky.
Like a vehicle, you'd think people should prove their competency and safety. This won't prevent mass shootings, but it'd be nice if there was something more in the way of people with no proper firearm training from buying one because they think it's a cool new toy.