I'm talking big issues like the recent vote by the British vote to exist EU. I'm talking federal bills/laws, does the US constitution allow these types of vote by the people?
The constitution starts (Art. 1 Sect. 1) with "All legislative Powers herein granted shall be vested in a Congress of the United States, which shall consist of a Senate and House of Representatives". Art. 1 Sect 7 says that Bills originate in the House or Senate (and revenue specifically in the House); and there is a procedure for presidential signing / veto / veto-override, in order to become law. Section 8 states what powers Congress has. In order for it to be binding at the national level for people to vote on some issue, there would have to be massive amendment of the Constitution, to give The People power to enact legislation, perhaps immune to presidential veto (or perhaps with absolute veto power, or some specified means of overriding a veto).
An alternative would be that for a special issue, Congress could "contingently" pass a bill, through the normal vote in both houses plus presidential approval, and the bill could then contain a further provision that the act is nullified if... and then you could specify some conditions. It could be for example a simple yes/no majority vote of those voters eligible to vote for president. In case Congress and POTUS approved the law but the majority of The People voted the law down, there is a distinct possibility that the constitutionality of the "refer it to a vote" provision would be ruled unconstitutional (conflicting with Art. 1 Sect. 1), and thus the bill would become law.