tl;dr I would say there was never a power vacuum. There was always at least someone nominally in charge of the state.

The King is still around through his execution in 1793; the Estates General meets in the spring of 1789- the group that will then transform itself into the National Assembly on June 17 of that year; a month later the same body will change its name to the National Constituent Assembly and begin radicalizing- that is moving from just looking to create an elected body to legislate with the King to seeking the creation of a new type of government altogether.

During the following year the Assembly will begin a series of government reforms, essentially replacing the old bureaucratic order with a new one. They also abolish the former division of society into estates and will gradually seek to standardize French society (creation of metric system, rational calendar, get everyone to speak French). By 1791, during the reforms, the Assembly will elect its first President and by the end of the year the Louis XVI, the French King will accept the Constitution that the Assembly had created.

The National Constituent Assembly now becomes the Legislative Assembly and the Revolution will continue to radicalize becoming by the end of the year the National Convention which declares the founding of the First French Republic. During this time the King will be tried and executed and the Committee of Public Safety (part of the Convention) will come to power. The Committee is extremely important for the revolution, because by this time (1792) France is under siege and assault by various European powers who want to stop the spread of revolution and who initially wanted to put the king back into power. The Revolutionaries themselves are interested in spreading their republican ideals across Europe (and the Mediterranean in fact).

The Convention will be the governing body of France until 1795 when the Executive Directory is declared and the more radical parts of the revolution are put down- by Napoleon. The Directory will rule over France until 1799 when Napoleon returns from attempting to spread the French Revolution in Egypt and leads a coup d'état against the Directory and replaces it with the French Consulate- in which he is a member in the ruling triumvirate.

Depending on how you define the French Revolution this is then the end and the beginning of Napoleonic France; or if you include that as basically just an extension of the Revolution; we can say Napoleon is in charge, being crowned Emperor in 1804 and ruling until his exile and the Restoration of the French monarchy.

So really; there was always someone in charge, someone issuing orders and making law, and generally they had full authority because they had the army backing them up.

Now; whether the people saw it as a successful revolution or not is tricky; if you mean the people in Paris, the commoners and etc; then immediately the revolution appeared successful, and in fact the Parisian crowds are going to push the revolution towards radicalism. Crowds will come watch the various assemblies shouting support to those they liked and shouting down the people they do not. There will be the mass publication of pamphlets and news papers in Paris keeping the reading public in the know. And when France is attacked many young Parisians will join the early revolutionary armies to defend the fatherland and before long the revolution will conscript tens of thousands from outside of Paris as well. So in Paris, the revolution was off to a rocking start and was both supported and driven by the people.

Outside of Paris- especially in areas loyal to the Crown and the Church- like say the Vendée; there was never any support of the revolution and in fact the revolution will at times turn into bloody civil war as the revolutionary government seeks to put down places like the Vendée, executing tens of thousands of people identified as counter-revolutionaries. Other parts of the country are more ambivalent or in fact support the revolution- especially when outside powers like Prussia try to invade France to put down the revolution- nothing like outside invasion to bring people together.

The fact that the revolution more or less begins with the meeting of the Estates General- an 'elected' body designed to address social grievances collected in the Cahier de doleances- grievances coming from all sections of society- many from the peasantry or the common people- meant that many supported the ideals of the revolution because they believed it was addressing the fundamental social inequalities that existed in French Society. Places like the Vendée only really rise in opposition when the revolution takes its radical turn and deposes the king and attacks the Catholic Church.

Hope this helps. A good account of the French Revolution that discusses much of this would be "A Concise History of the French Revolution" by Sylvia Neely; or William Doyles "The French Revolution; A Short Introduction"