You are correct to point out that a lack of being able to secure arms internationally stunted the Republican war effort, particularly in the areas of automatic weapons, artillery, and armor. However, unity being a strength of the Republicans isn't quite accurate. There was a good deal of infighting amongst the factions that made up the Republicans that detracted from their war effort.

The reasons the Nationalists won the Spanish Civil War are:

• Numerical superiority--throughout the war, the Nationalists simply had more troops than the Republicans
• Direct assistance from Germany and Italy--Germany gets a lot of notoriety for supplying the Nationalists with the Condor Legion, but the Italians supplied a much larger number of troops (and gave naval assistance as well)
• The Nationalists secured the main food-producing regions of the nation fairly early on, making the Republican efforts to feed their troops and citizens quite difficult
• Republican infighting took some troops away from the front and hindered the war effort in other ways
• Anticlerical violence by Republicans was used as a reason by foreign governments to not intervene in the war (this is less important than the above reasons, but was still a factor)

I listed the reasons in descending order of importance, based on my opinion after reading the various books available on the war. Beevor does the best job of the bunch among the writers of the one volume histories (Beevor, Preston, Thomas, Payne) when it comes to the military aspects of the war. Basically, having the numerical advantage allowed the Nationalists to take and keep the initiative throughout the war. With the mainland Spanish military forces being roughly split 50/50, the additions of outside troops along with troops from Spain's Moroccan territory (Regulares) gave the Nationalists a critical advantage in what turned out to be largely an infantry war. At nearly every turn the Nationalists were able to dictate when and where the fighting would take place and they could afford to take the casualties that offensive operations required. The few Republican offensives were met by the Nationalists shifting their forces to meet and defeat the Republicans, then returning to their previously planned advances. The Republicans were deliberately ground down and defeated.

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A quick note: be careful calling the Nationalists "fascists." The Falange--Spain's fascist party--got less than 0.1% of the vote in the 1936 elections that immediately preceded the outbreak of war. Out of nearly 9.5 million votes, the Falange didn't get 7 thousand. The Nationalists were a coalition of traditionalists, military supporters, two different types of monarchists, industrialists, large landholders, Catholic party members, and fascists. Franco was quite skillful in subsuming those various factions into his own. He was able to keep any one of them from becoming powerful enough to challenge him while at the same time keeping them unified enough to defeat the Republicans. So, calling the Nationalists "fascist" isn't quite accurate.

Followup questions from OP and others are always encouraged.