I don’t have sources in front of me, so I’m pulling this from memory from a college project long since passed.

Short Answer: The Spanish Civil War (1936-1939) destroyed the Spanish economy and nation to the point that the nation could not substantively contribute to either the Axis or Allied war effort.

Long Answer: They did, kinda...

The Franco regime that solidified power following the SCW closely resembled other nationalist governments such as Nazi Germany and Fascist Italy. Franco received material and military support from those two nations during the SCW and retained diplomatic ties with both following the end of the war. Hitler conducted talks with Franco in the early 1940s in part as a way to plan for eventually control of British Gibraltar.

The Spanish organized expeditionary forces that served with the Nazi’s on the Eastern Front. These “Blue Divisions” supposedly repaid the Dueda de Sangre (Blood Debt) that Spain incurred by receiving German military assistance during the Civil War. These units fought on the Eastern Front, most notably during the Siege of Leningrad.

Diplomatically Spain served as a thorn in the side of the Allies and Axis alike. A sort of quasi economic war erupted between the two factions over Spanish tungsten supplies. The Allies attempted to manipulate the market in such a way to either control or overprice the resource so to prevent the Axis from gaining it.

Adding to what’s already been said: The division azul consisted of a bit less than 50.000 soldiers in total. It was justified in the same way, as the Spanish Civil War before: Mainly as “holy crusade against the evils of Bolshevism”. Contrary to the official propaganda, the soldiers motivations were quite diverse. People joined for fighting communism, but also communists joined for joining the Soviet side when in Russia. Officers went for better prospects for promotion, criminals for escaping prison and among the soldiers there were some veterans of the civil war and the colonial war in northern Morocco. The most recent and detailed study on the topic is the following, sadly only available in German, as far as I know: Xosé M. Núñez-Seixas: Die spanische Blaue Division an der Ostfront 1941-1945. Zwischen Kriegserfahrung und Erinnerung, Münster 2016.

As already mentioned, francoist Spain’s overall “neutrality” was mainly forced upon it by the devastations of the civil war, ironically in many ways caused by the Italian and German intervention. This, however, didn’t prevent the regime from selling the neutrality, after the war was lost for the axis powers, as a simple result of “Franco’s genius”.