Yes, trade continued, although it was lessened.

From the late fifth century onward, the mass exploitation of natural resources ceases. The great mines in England and Switzerland shut, and are replaced by local, smaller-scale production. As the European population shrinks, we see less land under cultivation and forests starting to regrow (a process which begins to reverse itself in the late 7th c). By the end of the sixth century, the annona - the grain shipments from Egypt to Rome - have ceased, and by the end of the seventh century, the Mediterranean is no longer a Roman (or even a Christian) lake.

However, we still have a great deal of evidence for long-distance trade, particularly in luxury goods. Several Merovingian burial shrouds have been found embroidered with silk, for example. There is also substantial evidence for the continued trade in slaves, particularly to and from central Europe. The local artisan communities which spring up also become relatively wealthy. One such community of smiths located in the Jura (now Switzerland) was not only raising animals to be eaten at a relatively young age (a sure sign of wealth) but also receiving goods from across Gaul.

With respect to the use of roads, I'm hesitant to go into particular detail because the work I'm thinking of is still unpublished, but it should suffice to say that there is a slow transition away from roads and to the rivers as a means of transportation. This shift is mostly complete by the 8th century. However, the major roads not only continued to be used, but they still are in use today! If you look at this map and change the background layer to google streets, you can clearly see that many modern highways closely follow the paths of known Roman roads.

Some sources & further reading:

The definitive book on the early Medieval economy:

• McCormick, Michael. Origins of the European Economy: Communications and Commerce A.D. 300-900. Cambridge: Cambridge University Press, 2001.

One of the first discussions of Mediterranean trade in late Antiquity:

• Pirenne, Henri. Mohammed and Charlemagne. New York: Norton, 1939.

Pollen data for the regrowth of forest, etc:

• Rösch, Manfred. “Human Impact as Registered on the Pollen Record: Some Results from the Western Lake Constance Region, Southern Germany.” Vegitation History and Archaeobotany 1 (1992): 101–9.

• Gauthier, Emilie, and Hervé Richard. “La forêt jurassienne au cours des deux derniers millénaires à la lumière de quelques diagrammes polliniques.” In La mémoire des forêts: Actes du colloque Forêt, archéologie et environnement 14-16 Décembre 2004, edited by J.L. Dupouey, E. Dambrine, and C. Dardignac, 57–67. Nancy, 2007.

The archaeology of iron working in the 6-7th c. Jura:

• Fellner, Robert O., Ludwig Eschenlohr, Reto Marti, Maruska Federici-Schenardi, and Michel Guélat. Develier-Courtételle, un habitat rural mérovingien. Vol. 13–17. 6 vols. Cahiers d’archéologie jurassienne. Porrentruy: Office de la Culture :Société jurassienne d’Émulation, 2004.