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?
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.
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.