I'm not from the US and I don't have a lot of knowledge about US law, but it seems strange to me that the executive branch of the government is hosting petitions for what in most other countries would fall under the legislature.
Isn't it the task of Congress in the US to set policy about what requirements are attached to the expenditure of public money, does the executive branch really have any impact on stuff like this?
You have the philosophy right, but political reality has left that behind.
While the President doesn't have any official power over the legislating process, he can wield significant political pressure as de facto head of his party, via shaping public opinion from the bully pulpit and with the threat of a veto.
Presidents have, for some time now, been very active in setting/driving legislative priorities.
It would be possible in the US for something to be enacted by an Executive Action, basically a directive to agencies from the Whitehouse. There is also legislation at the moment in the House and Senata (FRPAA) that would also achieve the same thing. If there was Whitehouse support the chance of the legislation passing is higher and if the legislation gets support there is more chance of an Executive Action so this is going from both ends.
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.
Isn't it the task of Congress in the US to set policy about what requirements are attached to the expenditure of public money, does the executive branch really have any impact on stuff like this?