Congress could definately make this law, but they also delegate many policy details to the executive branch - so this is likely something that could be done by either branch right now.
Regarding the 'petitions'. The 'petitions' in question here have no legal standing at all. It is essentially a way for the current administration to let the people involved feel like their views are being heard. The end result of each of these is generally either a) a canned response about how they can't comment on the petition for whatever reason or b) a rehash of current policy.
The only real impact, if any, of the petition is if they sway the administration by demonstrating what public opinion on the matter is.
Regarding the 'petitions'. The 'petitions' in question here have no legal standing at all. It is essentially a way for the current administration to let the people involved feel like their views are being heard. The end result of each of these is generally either a) a canned response about how they can't comment on the petition for whatever reason or b) a rehash of current policy.
The only real impact, if any, of the petition is if they sway the administration by demonstrating what public opinion on the matter is.