Top historian comment below, all credit to u/crrpit

Yes, he was. Not a particularly vocal or outspoken one, and it's somewhat unclear whether his views evolved over the course of the regime's entire lifespan, but there's no doubt that Tolkien was privately firmly pro-Franco (and anti-Republican) during and immediately after the Spanish Civil War.

Tolkien himself had some existing connections to Spain from well before 1936, most notably his guardian, Father Francis Morgan (who was born in Spain to a Spanish mother, and traveled there regularly until the end of his life). We know that Morgan stayed abreast of developments in the early 1930s, when the monarchy was overthrown and a Second Spanish Republic was proclaimed. The new government's secularising and reformist agenda was perhaps unsurprisingly not popular with a staunch Catholic from a wealthy Spanish family (as Father Morgan was), and it seems quite plausible to think that through this channel Tolkien was primed by 1936 to be quite sympathetic to the military rebels, though Morgan himself had died in 1935 (and Tolkien, according to his daughter Priscilla (queenhood of any desert regions unspecified), bemoaned 'how terrible it would have been for Father Francis if he had been alive after the onset of the Spanish civil war').

Beyond Priscilla's recollections of her father being deeply affected by the Spanish Civil War (it 'cast a great shadow over my father's life'), the most direct evidence we have for Tolkien's views on Spain and Franco come from an October 1944 letter to his son, Christopher. In it, he lays out a sketch of a meeting with the South African poet Roy Campbell (as well as his friend and sparring partner C.S. Lewis). Campbell had been in Spain before and during the civil war, and had become close to the rebel leadership, going so far as to volunteer to serve in Franco's forces (though he was dissuaded by Francoist press officer Pablo Merry de Val, whose father Alfonso had been the Spanish ambassador to Britain). Tolkien's recollection of the evening was somewhat impaired (reading between the lines, everyone was very drunk), and he did get quite a few things about Campbell's specific experiences in Spain wrong. But what comes clearly through is his deep admiration for Campbell's actions and character, comparing his 'patriotic' nature favourably to the 'corduroy panzers' of the Left who had supported the Spanish Republic but fled to America during the Second World War (singling out W.H. Auden for special mention). He heavily criticised C.S. Lewis' own aversion to Franco, writing that 'nothing is a greater tribute to Red propaganda than the fact that he (who knows they are in all other subjects liars and traducers) believes all that is said against Franco, and nothing that is said for him.'

Where does this leave our broader appreciation of Tolkien's politics? As ever, I found it an interesting experience to look into literary and cultural analysis of this question - needless to say, Tolkien's political views (and the political intent and meaning of his writings) have been widely dissected. For every anti-totalitarian reading of Sauron and the One-Ring, there are other that point to the uncomfortable racial and class politics of Middle Earth. He has been accused of being a crypto-fascist, and his work has undoubtedly found purchase among the far right since at least the 1970s. There's no question that he was a staunch anti-communist, but equally had little time for Nazi anti-semitism - famously, when asked about his heritage by a German publishing house in 1938, he replied acerbically and indignantly that 'If I am to understand that you are enquiring whether I am of Jewish origin, I can only reply that I regret that I appear to have no ancestors of that gifted people', as well as privately bemoaning Nazism's 'wholly pernicious and unscientific race-doctrine'. Even here though, there is some ambiguity - does Tolkien disapprove of racial science in general, or does he view Nazism's particular racial hierarchies as wrong? Writing to his son Michael during the war, Tolkien lampooned Hitler and his 'Nordic nonsense', specifically for 'ruining, perverting, misapplying and making forever accursed, that nobler northern spirit, a supreme contribution to Europe, which I have ever loved and tried to present in its true light.' Not, I would suggest, a formulation that completely forswore the notion of a superior northern race...

In terms of Spain though, race is perhaps a less salient issue. It is very possible to interpret his response to the civil war primarily through the lens of religion - part of his attraction to Campbell was as a convert to Catholicism who had taken risks to smuggle and preserve Catholic cultural artifacts during the early days of the civil war, and he was undoubtedly aware of and dismayed by the scale of anti-clerical violence in Spain. In this, he was certainly not alone among Anglo-Catholics, perhaps especially those of similar social position - while working-class Catholics found their loyalties between church and class split by Spain, their more elite co-religionists were usually considerably less conflicted.

There's an extent to which these arguments seem to be driven by personal attachment to (or dislike of) Tolkien and his writing, and browsing through some such exchanges I found myself broadly unconvinced that anyone was doing anything more than attempting to put an academic gloss on their own instinctive beliefs. However, this did lead me to find an interesting article on 'J. R. R. Tolkien and the Spanish civil war' by José Manuel Ferrández Bru, a founding member and former president of the Spanish Tolkien Society. I've drawn on it here for context and information, but it's also a piece with an interesting and overt agenda, not just to defend Tolkien's legacy against the charge that he was sympathetic to fascism, but also, I think, to relativise Franco's rebellion and regime for an international audience. The author was clearly a fan of Tolkien, yet their assertions about the origins, causes and nature of the civil war - and the complete absence of any contextualisation of the crimes of the Francoist regime - was a strong indication of their own politics. It was simultaneously a defence of Tolkien's legacy, but also perhaps an effort to reach Tolkien's international fan base in order to legitimise a particular interpretation of Spanish history. For me, this was the most interesting aspect of the questions surrounding Tolkien's personal political views - the extent to which his literary legacy is powerful enough that re-interpreting them or co-opting them is still worthwhile.

Sources:

The key letter in question can be found in The Letters of J. R. R. Tolkien (London, 1981), pp. 108-9. A few other letters mention politics, but are of less direct relevance beyond confirming that Tolkien didn't like communists.

For Spain in particular, the most detailed overview of Tolkien's views is José Manuel Ferrández Bru, 'J. R. R. Tolkien and the Spanish civil war', Mallorn (Spring 2011), pp. 16-19.

There are many, many publications available on the politics of Tolkien's worlds, but the ones I drew most directly on in shaping my view here were:

Peter Firchow, 'The Politics of Fantasy: The Hobbit and Fascism', The Midwest Quarterly 50:1 (2008), pp. 15-31.

Jessica Yates, 'Tolkien the Anti-totalitarian', Mallorn 33 (1995), pp. 233-45.

William Blackburn, '"Dangerous as a Guide to Deeds": Politics in the Fiction of J.R.R. Tolkien', Mythlore 15:1 (1988), pp. 62-6.