Certainly, there was extensive ancient Egyptian maritime trade in the Aegean and East Mediterranean, particularly during the Late Bronze Age. This period saw a massive increase in international trade, with trade routes heading from East-to-West, following the coast along Egypt, the Levant, southern Turkey and the Aegean and back. During this time we see the development of complex markets, standardised trade tools such as 'Canaanite' jars and ox-hide ingots.

There is a lot of archaeological evidence or Egyptian objects in Aegean contexts, particularly in the early part of the Late Bronze Age, when the Egyptians were most dominant traders in the region along with the Minoans on Crete. Later, when Cyprus and Syria-Palestine became players in international trade, Egypt lost some of its dominance. Nevertheless, there is evidence for Egyptian goods all around the East Mediterranean. (Cline, Sailing the Wine-dark Sea). Shipwrecks such as Ulu Burun and Cape Gelidonya also provide a good insight into the types of goods traded: precious metals, nuts, grains and fruit, unworked ivory, manufactured goods like pottery, jewellery, cylinder seals, bronze tools and weapons, faience and amber beads, myrrh or frankincense (Bass - A Bronze Age Shipwreck at Ulu Burun; Haldane - Direct Evidence for organic cargoes in the LBA).

There are also examples in texts for maritime trade. As an example, letters between Egypt and the king of Alashiya (Cyprus) show that trade relations could be carried out by kings as a kind of gift-giving ceremony. These letters often provide an inventory of the goods being traded (Moran - Amarna letters).

There is also plenty of evidence to suggest that the Navy formed an important part of Egyptian Asiatic campaigns. We have the account of a naval officer who took part in the expulsion of the Hyksos just before the New Kingdom (Pritchard ANET p233). There's also some reliefs at a mortuary temple of Hatshepsut that give an account of the naval expedition to Punt. The Gebel Barkal stela shows that ships were built in the coastal city of Byblos and sent to Egypt.

So yes, they were heavily involved in trade around the East Mediterranean and the Aegean, and they had a strong naval presence as well.