It depends on what you think caused the crash, and I'm going to focus on the Depression as a whole rather than specifically on the equities crash. Friedman has a very prominent, monetarist argument that the Fed's decision to tighten monetary policy (raising rates) led to a cash crunch, which led to the crash and the depression. The Fed is ostensibly independent of the Federal Government, but could have been influenced by the appointments made by the Republican led government of so long.

This is not the only potential cause. Keynes argued that the Depression was caused by a significant fall in aggregate demand that was not made up by the Federal government soon enough. The time between the fiscal stimulus of FDR and the beginning of the economic depression led to the prolonged contraction that was only really made up with the increased war spending during WWII.

There are more arguments that prolonged periods of deflation led to increasing debt burdens and pessimistic outlooks on the economy, which fed a vicious cycle.

There are further arguments that there was a failure to control public expectations about the economy (regardless of the issues mentioned above), which led to pessimism, cash hoarding, and reduced consumption and investment, which caused the contraction.

Modern mainstream opinion for the cause of the Depression integrates the above reasons. You can see how they fit together fairly well. There are heterodox theories (e.g. Austrians blame credit expansion, Marxists blame inherent failure in the capitalist model, etc.). However, most explanations for the Depression put a significant amount of blame on the government for tightening monetary policy and not making up for discrepancies in demand by injecting liquidity or stimulus (not even mentioning other issues like passing the Smoot-Hawley Tariff, which had a non-zero impact on the economy). The actions of the government are not solely to blame for the Depression, obviously, but they greatly exacerbated it. Thus, you can say that the universal Republican control of the Federal government was connected to the Depression.

It's certainly connected. But the size and direction of effect would require a lot of research to figure out. An easy connection to make between this Republican majority and the broader picture of the great depression are increasing tariffs.For example, the Republican-backed Smoot-Hawley Tariff Act increased import duties by 20% and resulted in a 22% decline in US imports during the two years after it was passed in 1930 (slightly after your time frame, but by the same congress). This Act contributed towards the decline from recession to depression.^1

If you want an example before the crash, the Revenue Act of 1928 lowered the tax on corporate income from 13.5% to 12%. It also lowered several commodity taxes. The total effect of the act was a $222,222,495 reduction in state revenue.^2 I'm not nearly qualified to try and explain the specific effects of this act. But it makes for a good example of how congress directly effected the economy during this time.

Before drawing too many comparisons to the past, keep in mind that the political parties are fundamentally different now than they were in the interwar period. You can't directly connect the Republican party of today to the Republican party of 1928. The fact that Republicans won the senate race in California and the Democrats won the seat in Texas that year helps drive that point home.

1: Irwin, D. A. (1998). The Smoot-Hawley Tariff: A Quantitative Assessment. The Review of Economics and Statistics, (2). 326.
2:Blakey, R. G. (1928). THE REVENUE ACT OF 1928. American Economic Review, 18(3), 428.

None of the posts here actually answer the question, which is very simple: was the total control of the government by the Republican Party and/or any ensuing policies responsible for the Great Depression?

To properly analyze this, we need to look at the parts of the government where the Republican administration or Republican party members had significant control of government policy as well as the political will to utilize it. The US government of the 1920s had significant existing institutions that would act in certain ways to amplify or damper the effects of the Great Depression, but that is extraneous from the question itself, which is whether the GOP is actually responsible. As such I will discount the role of the Federal Reserve, as the Federal Reserve was for all practical purposes independent from the actual ruling party. Keep in mind that the Board of Governors for the Federal Reserve serve 14 year terms, and that the Chair of the Federal Reserve in 1929, Edmund Platt, was appointed in 1920. In any case, /u/elev57 does an adequate job giving a brief summary of the general economic thought re: the Federal Reserve's actions during this period, or lack thereof.

Thus, the main action that we will look at during this period is the Smoot-Hawley Tariff. This tariff was initiated by Republicans, passed by Republicans, and signed into law by a Republican, with a Republican court at the ready to defend it from court challenges. It has been brought up multiple times as a contender for contributing to the Great Depression. But, what exactly was its contribution?

The Smoot-Hawley tariff, to give a brief background, was instituted as the global economic slowdown of the late 1920s began to take shape. One major priority was the protection of American farmers. Due to increasing mechanization of agriculture, agricultural yields were booming and food supply grew. At the same time, there were also products being exported to the US. This apparent supply glut resulted in lower prices, which concerned US farmers.

The tariff was thus proposed initially as a way to protect US agriculture. Not only were Smoot and Hawley attempting to appeal to their voters, but the Republican party had made a promise as part of Hoover's campaign to do so, not to mention the obvious strategic necessity of protecting domestic food production and infrastructure. However, the tariff quickly swelled to include other industries, including those of manufactured products, as well as other industries backed by individual Senators. Douglas Irwin noted in his "Peddling Protectionism" that the bill essentially became a pork bill to protect pet industries backing various supporters. Hoover actually opposed the bill, but was forced to give in due to internal party pressure.

Needless to say the results were about as expected. Other nations enacted retaliatory tariffs, which in combination with the actual economic crisis resulted in a collapse of trade and deepened the effects of the Depression. Yet Irwin is cautious to attribute the tariff itself to the Depression itself-and for good reason: because while trade and monetary policy both concern economics, the two are often separate when it comes to public policy. It certainly exacerbated the effects, but a fair comparison would be someone getting bronchitis while suffering from a cold-the bronchitis would not have been able to spread without the cold, and it certainly didn't cause the cold, but it certainly makes the patient feel much more ill.

Sources:

Douglas A Irwin: Peddling Protectionism

Further Reading: Thomas Ferguson's "From Normalcy to New Deal: industrial structure, party competition, and American public policy in the Great Depression" gives a model on how this sort of political development was possible in the US government of the 1920s.