This is a graphic of the U.S. Army Infantry Regiment (Medium Tanks) in 1939 based on The Nafziger Collection's TO&Es. There was only one medium tank regiment at the time, the 67th Infantry Regiment (Medium Tanks). However, for basically the entire interwar this was a small formation.
Prior to the 67th Infantry Regiment's designation on 31 October 1932, it was the 2d Tank Regiment (Heavy) with only one active battalion (2d Battalion, which before 1929 was the 17th Tank Bn. (Heavy)). On 15 September 1931, almost all of 2d Battalion was inactivated, leaving only Company F. On 4 April 1936 the regiment's until then organized HQ was organized as an inactive unit with organized reserve officers, but for most of the 1930s the 67th Infantry was just its Company F at Camp Meade. On 1 October 1939, after the German invasion of Poland, 2d Battalion HQ and Company D were reactivated. On 5 June 1940, after the German invasion of France, the entire regiment was activated at Fort Benning. One month later on 15 July 1940, with the formation of the Armored Force as a quasi-branch to end the infantry/cavalry tribalism holding back combined arms integration, 67th Infantry Regiment (Medium Tanks) was redesignated as 67th Armored Regiment (Medium) and assigned to the new 2d Armored Division. As 2d Armored Division remained a "heavy" division with 2 Armored Regiments instead of 3 independent Tank Battalions, the 67th Armored Regiment continued to exist throughout the war.
Before the U.S. Army brought tanks under a unified Armored Force in 1940, it basically consisted for four main unit types split between the Infantry Branch (tanks) and Cavalry Branch (combat cars):
66th Infantry Regiment (Light Tanks), with a full amount of tank battalions but dispersed and without a real RHQ. There were also the 68th and 69th Infantry Regiments (Light Tanks), but these were inactive with organized reserve personnel.
67th Infantry Regiment (Medium Tanks), although only Company F was active the entire interwar.
Divisional Light Tank Companies, 8 Active (1st to 7th Divisions + Hawaiian Division) and 18 National Guard (although most NG ones only had a 2 tanks for training out of an 18-tank paper strength).
Even the Cavalry Divisions had inactive (organized reserve) Tank Companies as opposed to troops which were under the infantry. Except for the 7th Cavalry Brigade at Fort Knox, these infantry tanks were the extent of the armor in cavalry formations even if they were purely cadre units
7th Cavalry Brigade (Mechanized), which in theory was meant to have 112 combat cars (the cavalry term for light tanks as "tanks" were the legal responsibility of the Infantry Branch).
Going into World War II, this is what was done with most of the interwar tank units:
66th Infantry (Light Tanks), 67th Infantry (Medium Tanks), and 68th Infantry (Light Tanks) [which was formed by redesignating divisional Tank Companies 1st to 7th] were brought under a Provisional Tank Brigade at Fort Benning in 1940. This formed the core of the 2d Armored Division, from which point they became 66th, 67th and 68th Armored Regiments. The 66th and 67th Armored stayed with the 2nd Division, but the 68th was broken up to form the independent battalions of the 6th Armored Division.,
HHT, 7th Cavalry Brigade (Mecz.) became HHC, 1st Armored Division. The brigade's 1st and 13th Cavalry Regiments (Mecz) initially became the division's 1st and 13th Armored Regiment. The 69th Infantry Regiment (Light Tanks), an inactive unit out of Minneapolis during the interwar, initially became the division's 69th Armored before being transferred, inactivated, reactivated, and then split up to form 69th Tank Bn, 708th Tank Bn, and Troop E, 86th Cav Recon Sqdn.
Divisional Tank Companies mostly went to separate Tank Battalions in the GHQ reserve or Tank Destroyer companies in the National Guard and the Hawaiian Divisions or the 68th Infantry Regiment (Light Tanks) in the Regular Army:
1st — 7th Tank Companies (Regular) — Formed elements of the 68th Infantry Regiment (Light Tanks) on 1 January 1940. These became Companies A, B, C, D, E, and F (5th and 6th Tank Companies were merged into a single Company C).
11th Tank Company (Hawaiian Division) — Combined with a cadre from the 193rd Tank Battalion to form the 762d and 763d Tank Battalions at Schofield, HI on 23 April 1942. The 193rd Tank Battalion continued to exist and ended the war in Okinawa.
27th (New York), 26th (Massachusetts), 29th (Virginia), 43d (Connecticut) Tank Companies — Formed the 191st Tank Battalion on 20 February 1941.
32nd (Wisconsin), 33rd (Illinois), 37th (Ohio) and 38th (Kentucky) Tank Companies — Formed the 192d Tank Battalion on 20 December 1940. It was sent to the Philippines because it was combat ready and lost at Bataan in April 1942.
30th (Georgia NG), 31st (Alabama NG), 36th (Texas NG), and 45th (Colorado NG) Tank Companies — Formed the 193d Tank Battalion on 21 January 1941
34th (Minnesota), 35th (Montana), 40th (California) Tank Companies — Formed the 194th Tank Battalion on 12 March 1941. Except for Company B (former 35th), it was sent to the Philippines because it was combat ready and lost at Bataan in April 1942.
44th Tank Company (New Jersey) — Seemingly disbanded around the time the division was inducted into federal service in September 1940, with personnel absorbed into other divisional formations. I haven't found anything more detailed than that though. In the Executive Order ordering National Guard units into active service the Tank Companies are notably excluded
41st Tank Company (WA) — Reorganized as Company C, 103d Antitank Battalion on 17 September 1940. On 15 December 1941 this became Company C, 803d Tank Destroyer Battalion as that function passed from the infantry branch to tank destroyers (page D/6)
28th Tank Company (PA) — Haven't found what happened to it. It was seen on exercise in summer 1940 in New York (page 7). I suspect it would have been reorganized or disbanded around the time the 28th Division was called into federal service in 17 February 1941
Sources:
Cameron, Robert S. "Mobility, shock, and firepower : the emergence of the U.S. Army’s armor branch". A section on Infantry Tank Organization in the 1930s is from pages 127 to 132
Clay, Steven's "U.S. Army Order of Battle 1919–1941" series, particularly Volume 1 (Major Commands and Infantry Organizations) and Volume 2 (Cavalry, Field Artillery, and Coast Artillery, 1919–41)
Stanton, Shelby. "Order of Battle U.S. Army, World War II". Armored Regiments start on page 286 and independent Tank Battalions start page 296
Nafziger's digitization of TO&Es, 1939-1940. Ctrl+F "Medium Tank " to find the relevent ones