Probably not.

First, it's difficult to find what Alf Landon really said. The claim that he advocated for the US to join the EEC comes from his New York Times obituary, quoted below:

He argued that the United States should join Europe's Common Market in 1961, when President Kennedy urged only cooperation.

We really don't know what he actually argued, or in what form. In a lecture Landon gave at Kansas State University, Landon said (emphasis my own):

In October 1961 when the White House was divided on whether to support the fledgling European Economic Community, or whether to request a year's extension of the Reciprocal Trade Agreements Act I urged support of the E.E.C. as the most realistic step toward economic anti-political stability, and hence world peace. Why? Because the E.E.C. was founded on the simple principle of removing nationalist barriers through international trade.

This gives us some additional context; Landon was pro-EEC, but it doesn't sound like he advocated for the US to join, he just believed the US should support the EEC bloc vis-a-vis tariff reductions and trade liberalization. I haven't been able to find out what he exactly said, and to whom he said it; the Kansas Historical Society maintains an archive of Landon's papers, but they are not digitized.

The US State Department maintains a wealth of documents related to previous administrations' foreign policy online, including diplomatic communications and memoranda of conversations between US and foreign officials. During the Kennedy administration, the development of the EEC's Common Agricultural Policy and its impact on US access to European markets was a common, prevalent topic of discussion. US policymakers, including Secretary of Agriculture Orville Freeman had concerns that the CAP would be protectionist in nature and would exclude American agricultural exports from the European market. When EEC President Walter Hallstein visited Washington DC, he held a separate meeting with Freeman to discuss the CAP, and President Kennedy addressed the CAP and US access to the EEC in communications to Congressional leaders and bilateral communiques to Konrad Adenauer, the West German Chancellor. As time progressed, the US and the EEC remained at loggerheads with respect to the CAP. When Freeman traveled to Paris to attend a summit of OECD agriculture ministers, he noted in a memo to Kennedy on his return that:

The French are intractable. The French Agricultural Minister is the strongest personality of the Agricultural Ministers and as such will exert, even if French agriculture was not generally speaking dominant, great influence.

and that:

The European Nations will not accord any trade opportunities to American agriculture unless vigorous coordinated consistent pressure is brought to bear. Without attempting to evaluate each nation’s posture, it is clear that in both Germany and France prominent and effective political leadership is very responsive to what is felt to be national interest and as such not concerned with the rights or interests of the United States... The fact that we have contributed enormously to their present strong economic position and also to the furtherance of the Community as a practical matter has almost no influence. It makes nice drawing room conversation, but it is of little practical use at the bargaining table.

At the summit, Freeman also raised concerns about the CAP's compliance with the General Agreement on Tariffs and Trade (the GATT), the main postwar international agreement that governed global trade liberalization.

In addition to disputes over agricultural products, there was tension between the US and Belgium over tariffs on textile and glass imports, which caused significant consequences in Belgium. In a diplomatic note to the US ambassador to Belgium in response to the tariff increases, the Belgian Foreign and Commerce minister stated that the incident prompted him to no longer consider Belgium's relationship with the US a partnership of equals.

These disputes fit into the larger discussion of trade liberalization and disputes Landon mentioned in his lecture. Given this context, it is fair to assume that Landon did not advocate the US to join the EEC and by extension the Common Market, as his New York Times obit states, but instead advocated removing trade barriers and liberalizing tariffs, which he believed would promote global peace.

Now, could the US even join the EEC in the early '60s? No. The EEC's expansion priorities primarily involved members of the European Free Trade Association (the EFTA), another trade bloc that emerged after WWII (the EFTA still exists; its members include Norway, Iceland, Switzerland, and Liechtenstein). The US was involved with this discussion as well; Hallstein discussed British accession to the EEC with President Kennedy during his visit to Washington in 1961 and there is extensive communication between the State Department in Washington and US Embassies across Europe concerning the subject. Between ongoing trade disputes and other expansion priorities, it is highly unlikely US accession was even on the table for either EEC or member-state policymakers.