The largest pre-war French far-right party was the PSF (French Social Party). This was the political outgrowth of the Croix-de-Feu paramilitary league, which was banned by the government in 1936. While it was formed too late to gain any real electoral traction in the 1936 elections (only six members elected) it grew increasingly influential across the Right during Popular Front period. It almost certainly would have done well had war not postponed further elections.

However there is some debate as to whether the PSF was fascist or merely extreme-right. Either way, it didn't thrive under Vichy. Like all Third Republic parties, there was no room for the PSF in the anti-parliamentary Vichy regime and attempts to rebuild the Croix-de-Feu stumbled in the face of opposition from the official regime-sanctioned Legion of Veterans. Personal clashes kept its leader, La Rocque, away from the levers of power in Vichy and he was eventually arrested by the Germans.

There is, however, no doubt about the politics of Jacques Doriot's French Popular Party (PPF), which was explicitly fascist and enthusiastically collaborationist. It was also the largest fascist group active in France during the war. Doriot was an interesting and charismatic character who enjoyed the patronage of the German envoy to Vichy (Otto Abetz). But he too never got near power. Far from openly favouring the fascists, the Germans preferred to use them as a threat against the regime in Vichy. It was only in the closing years that PPF members came to play a real role in France, eg Darnard's Milice. The same can be said about the rest of the explicitly fascist fringe groups and writers (eg Brasillach and Celine) that hung around Paris during the Occupation.

Vichy itself, insofar as it was at all ideological, remained an authoritarian and traditionalist regime. The fascists were held in contempt by Petain and the conservative military. The actual ideological composition of Vichy is an entirely different topic (and was arguably close to fascism itself) but few self-proclaimed fascists got near executive power in Vichy.

Sources: Robert Paxton's Vichy France is old but, for me, probably still the best one volume examination of the Vichy regime. It's rivalled by Julian Jackson's France: The Dark Years; Jackson's The Popular Front in France is also a superb introduction to French politics of the 1930s. Rene Remond's The Right Wing in France is also old, and doesn't spend a huge amount of time on fascism, but is a great survey of the French Right since the Revolution.

There were a couple of militaristic/antisemitic parties and movements.

A lot of that type of right-wing thought came out of the Dreyfus affair, which is pretty much the defining politic event of Third Republic/fin-de-siecle France. Antidreyfusards were the far right.

Before the Dreyfus affair, there was an earlier rightwing movement was boulangerisme. It was based around the popular general Georges Boulanger, and incorporated a lot of elements that we might call protofacist - demagoguery, military parades, strong nationalism etc.

After Dreyfus, there were a few movements, but most were united around antisemitism. There was the prominent writer and ideologue Maurice Barres, and a lot of right-wing thought in the catholic press.

The party closest to what you're describing, though, is Action Francaise, who leapt to prominence after the Dreyfus affair, particularly through the actions of Charles Maurras. They were nationalist and antisemtic, although not as typically fascist as equivalent parties in neighbouring countries. They were very supportive of Petain's government, as they thought he was an instrument of national unity, and it seems that Maurras even half-believed in the fiction of Vichy France.

Some good sources on the subject:

Articles

Arendt, H., ‘From the Dreyfus Affair to France today’ in JewishHistorical Studies, iv:iii (1942), pp 195-240

Byrnes, R. , ‘Antisemitism in France before the Dreyfus Affair’ in Jewish Historical Studies, xi:I (1949), pp 49-68.

Fitch, N., ‘Mass culture, mass parliamentary politics, and modern anti-Semitism: The Dreyfus Affair in rural France’ in The American Historical Review, xcvii:i (1992), pp 55-95

Green, N., ‘The Dreyfus Affair and ruling class cohesion’ in Science and Society, xliii:I (1979), pp 29-50

Volkov, S., ‘Antisemitism as a cultural code: reflections on the history and historiography of antisemitism in Imperial Germany’ in Leo Baeck Institute Yearbook, xxiii:i (1978), pp 25-46

Books

Chapman, G., The Dreyfus case: a reassessment (London, 1955)

Datta, V., Heroes and legends of fin-de-siècle France (Cambridge, 2011)

Drake, D. French intellectuals and politics from the Dreyfus Affair to the occupation (Houndmills, 2005)

Fortescue, W., Third Republic in France: conflicts and continuities (London, 2000)

Kaplan, Z. J., Between the devil and the deep blue sea: French Jewry and the problem of church and state (Atlanta, 2009)

Kedward, R., La vie en bleu: France and the French since 1900 (London, 2005)

Lewis, D. L., Prisoners of honor: the Dreyfus affair (New York, 1994)

Lindemann, A. S., The Jew accused: three anti-semitic affairs (Dreyfus, Beilis, Frank) 1894-1915 (Cambridge, 1991)

McClelland, J. S. (ed.), The French Right from de Maistre to Maurras (London, 1970)

Rémond, R., The Right Wing in France from 1815 to de Gaulle (Philadelphia, 1969)

Saunders, D., ‘France on the knife-edge of religion: commemorating the centenary of the law of 9 December 1905 on the separation of church and state’ in Levey, G. B. and Modood. T. (eds) Secularism, religion and multicultural citizenship (Cambridge, 2009), pp 56-81

Steiman, L. B., Paths to genocide: antisemitism in Western History (Houndmills, 1998)

Tannenbaum, E. R., The Action Française: die-hard reactionaries in twentieth-century France (New York, 1962)

Weber, E., Action Française: royalism and reaction in twentieth-century France (Stanford, 1962)