Did the Axis engage in trade with other countries during the war?

Germany

Germany's international trade was largely restricted to overland routes due to the allied blockade. In 1942, Germany's main exports consisted of engineering products, metals and fuels. In addition to trading with the countries it occupied, Germany imported tungsten from Spain and chromite from Turkey.

Due to the Skagerrak blockade (pdf), Sweden became heavily dependent on Germany as a trading partner, importing fertilizers and coal (among other items) and exporting iron ore, ball bearings and wood. Switzerland traded currency and gold with Germany as well as precision machine tools, watches and other items.

Before the invasion of the Soviet Union, Germany obtained raw materials (pdf), including rubber, from Southeast Asia via the Trans-Siberian railway.

Italy

Prior to the war, 80% of Italy's trade came through the Straits of Gibraltar. Italy's ability to trade was also severely hampered by a lack of foreign currency and limited industrial productivity. Germany was a major trading partner, and Italy also traded with neutral countries such as Spain and Switzerland, and even sold planes to Sweden.

Japan

Like Italy, Japan also had limited foreign currency, and was also hampered by an inadequate merchant fleet. Prior to Pearl Harbor, the US was a major trade partner despite increasing restrictions imposed by the Americans.

Even more than Germany, Japan exploited its occupied territories ruthlessly, partly under the guise of the Greater East Asia Co-Prosperity Sphere. Indonesia became the main source of oil for Japan., though this was still insufficient. More tin and rubber than was needed came from Malaya, especially after Operation Barbarossa effectively ended the possibility of exporting any significant surplus production by land to Germany. Among other products, rice was obtained from Thailand, with iron and copper amongst those from the Philippines.

Trade between the Axis powers

There were obvious difficulties in shipping goods from Japanese controlled areas in Southeast Asia all the way to Europe. Nonetheless, blockade-runners

between 1941 and 1944....delivered 43,983 tons of natural rubber to the German and Italian war industries. They also carried 68,117 tons of other essential materials, mostly from Southeast Asia, such as tungsten, tin, and quinine, and altogether about two-thirds of the German annual requirement for these items. ... Although by mid 1942 Germany and Italy had unlimited access, at least in theory, to natural rubber, shipping it safely to Europe became extremely dangerous.... While the only viable route was now via the sea, the Allies’ blockade became so effective, especially after the introduction of the Checkmate System on 8 June 1943, that fewer and fewer Axis blockade-runners succeeded in reaching Europe. By late 1942 and early 1943, only one of the six ships that left for Europe reached its destination.

In return, Japan received military technology, though the Germans actually had little to spare. With so few surface ships making it (pdf),

In summer 1944 Hitler forbade the employment of German surface blockade runner because of the risks....The last German surface blockade runner ship reached France in November 1943

Thus, submarines were increasingly used despite their limited capacity (up to 160 tons), and many never reached their destination:

Among the tens of German and Italian submarines that were sent to the Indian Ocean and the four Japanese ones that left for Europe, only a fraction succeeded in making their way back or surviving the war. In 1944 alone, nine of the twelve submarines that left for Europe were sunk or forced to return.

During 1944-45,

these submarines carried 2,606 tons of vital raw materials (mainly tin, rubber, tungsten, quinine, and opium, in descending order) from Asia to Europe and 2,070 tons (made up of mercury, lead, aluminium, glass, and steel) in the opposite direction.

but...

due to the high ratio of losses en route...no more than 611 tons of materiel (23.4% of the total carried) arrived in Europe – considerably less, ironically, than the materiel (869 tons or 42% of the total) which managed to arrive in Asia