Is there any historical precedent for two countries being at war and agreeing to allow some commerce to continue?

Note: This answer has been revised and extended from its original version.

There is not only historical precedent for countries at war still doing commerce with each other, it seems to have happened a lot more often than one might think. Barbieri and Levy write in Sleeping with the Enemy: The Impact of War on Trade:

Contrary to both liberal and realist theories of interdependence and war, however, there are numerous historical cases of trading with the enemy during wartime, including trade in strategic goods that directly affect the ability of a state to prosecute the war. This is quite evident from numerous historical accounts (Giltner, 1997; Levy, 1998a). For example, the Baltic trade was so essential to the economy of the Netherlands in their Eighty Years’ War with Spain (1565–1648) that the Dutch served as carriers of naval stores for the Spanish. In this way the Dutch earned monies to pay the forces that protected Dutch frontiers against Spanish attack, while Spain secured the stores that helped maintain its fleets engaged in the protection of Spanish commerce against Dutch attacks (Howard, 1976:44).

Or consider the Anglo-Dutch Wars, in which British insurance companies continued to insure French naval and commercial ships and to pay enormous sums to replace ships that were actively being searched and destroyed by British warships (Pares, 1963). Trading with the enemy was also widespread during the Seven Years’ War, the War of 1812, and the Crimean War (Levy, 1998a), and this phenomenon has not ceased in this century.

Grinberg writes in Wartime Commercial Policy and Trade between Enemies:

Conventional wisdom suggests that trade is the first casualty of war. Because the gains from trade can be converted into military capabilities, trading with the enemy is akin to selling the opponent the gun they will use to shoot you. The empirical record of wartime trade, however, suggests otherwise. For example, World War I, a total war in which the majority of the states involved fought for their very survival, saw extensive trade between enemy belligerents. Britain continued to trade with its enemies until October 1, 1918 — one month and eleven days before the Armistice. In fact, Britain started the war with restrictions on the export of only 20 percent of the goods that it ultimately prohibited from reaching the enemy. Even after a year of fighting, by the end of August 1915, around half of the products that would eventually be prohibited were still allowed to be legally traded with enemy states.

World War I is hardly unique in that trade occurred and varied during the war. Some enemies continue to trade throughout the war—for example, India and Pakistan in the First Kashmir War (1947–49) and Yugoslavia and Croatia in the War of Bosnian Independence (1992). Other states sever trade immediately at the start of the war—for example, England and Argentina in the Falkland Islands War (1982) and India and Pakistan in the Kargil War (1999). Yet, other states start off trading with the enemy, only to change course during the war, as occurred, for example, between Ethiopia and Somalia in the Second Ogaden War (1977–78). There is remarkable variation in wartime trading patterns between adversaries.

The latter article seems to be an abridged version of the author's dissertation (pdf), which contains further examples:

For the first six months of [WWI], raw materials such as hemp, flax, and tallow were allowed to be exported [from Britain] to Germany and licenses were granted to merchants to import hosiery needles and aniline dies in return. During World War II, likewise an existential struggle between Britain and Germany, Britain allowed German dyestuffs to be imported to keep its textile industry alive. On the other side of the globe, while the Japanese Army was slowly conquering parts of China, the free parts of China continued to trade with Japan, much to the ire of their Western supporters. During the War in the Donbass (2014), Ukraine continued to import Russian coal.

Especially regarding trade between Britain and Germany in WWI, this might seem to conflict with the Blockade of Germany. But this blockade didn't start full-bore with the beginning of the war. Britain had opted for a "distant blockade", blocking not only entrance to enemy harbors from within the three-mile limit, but basically blocking entrance to the English Channel and the North Sea as a whole. This version was of unclear legality, and neutral countries, among them the US, weren't exactly happy about it. Also, the British business community was used to the pre-war "laissez-faire" style of things and didn't care much for restrictions and regulations.

So, the blockade started with a restricted list of contrabands (goods that would be confiscated) and restricted authority for the blockading ships. Over the course of the war, the blockade would be tightened more and more.

For more information on this topic, see for example British Blockade of Germany in WW1, The British Blockade During World War I: The Weapon of Deprivation and Naval Blockade (of Germany).

A more aggregated view can be gained from a paper by Glick and Taylor (Collateral Damage: Trade disruption and the economic impart of war, 2005). It looks at the costs of war for 172 countries between 1870 and 1997. The authors built a statistical model from that dataset. The math involved is way beyond the scope of this question, but Figure 1 (page 53) is interesting. The figure shows the "impact of war on trade for a given country pair" (from the dataset). While "at war", "trade between adversaries" declines by 85%. This means, while the trade declines quite steeply, there still is 15% trade compared to the pre-war level.

A more anedoctal point is the exchange of rubber for optical glass between Britain and Germany during WWI. In short: Britain needed optical glas for binoculars, gunsights and so forth. Germany needed rubber. The British Ministry of Munitions proposed an exchange of goods through intermediaries in Switzerland. There are different accounts on whether or not any goods actually were exchanged before both sides found "better sources" for their needs at a later stage.