To take a slightly different tack on answering the question - you ask about what happened to the term. I'm happy to report that the noun "víkingr" and the phrase "í víking" remained in common use in Old Norse for centuries after the end of the so-called "Viking Age".

There are occasional appearances of the word in the runic corpus, either as an ordinary word or, as in the case of Vg. 17, a runestone from the island of Torsö in modern Sweden, a name. However, the vast majority of the instances of the name are from the sagas, prose texts created mostly (but not exclusively) in Iceland out of oral traditions in the late 1100s through the 1300s. the dictionary of Old Norse Prose records 118 instances of the noun "víkingr" in widely disparate texts, from homilies to royal sagas like Sverris saga (composed in the early 13th century and probably commissioned by King Sverrir of Norway himself) to annals (the annales regii claim there was a fight between a King Olaf, probably Olaf of Sweden, and the Sota Vikings in 1007). Some of these instances are not stories derived from the Viking Age - one attestation calls the birkibeinar, a faction in the 12th and 13th century Norwegian civil wars, "vikingar" as they flee into the woods to avoid a battle they were going to lose. So there, it's probably not referring to them doing naval raids, but instead roughly equivalent to "brigands".

the phrase "vera/liggja/fara í víking" appears somewhat rarer, being listed 19 times in the corpus covered by the dictionary (which I use as it lists the location and offers a hyperlink to every attestation). however, a similar diversity of locations is attested, occurring in texts of a wide generic and temporal distribution. While these texts are largely stories about the Viking Age, it appears once in Sturlunga saga, a compilation of sagas about the period 1180-1280, give or take. This suggests that there was at least some kind of comparable practice of seafaring that was preserved, though the context of the saga at that point is not nearly clear enough to strongly identify what that practice actually is and how it differed from Viking-Age naval activities.

Nevertheless, the linguistic attestations make it more than clear that the term remained in use well into the 14th century, largely to refer to the same act of sailing and raiding. It appears mostly as a story-telling term, a thing that people used to do, but there are some hints of a conceptually similar practice continuing long, long after western Europe stopped having to resist raids and invasions from Scandinavia.

While there will always more to be said, I hope these of my previous answers might be useful for OP's question to clarify the situations surrounding the alleged end of the concept of the Viking Age (s):

• At What Point Were Vikings no Longer Considered Vikings, and Just Seen as Christian/Christianized Danes, Swedes, Norwegians, etc.?
• How active were baltic raiders prior to the Livonian and Teutonic orders arrival?
• Has Norway invaded England since 1066

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In short, the Viking Age(s), at least in English, is primarily rather a historiographical concept of periodization, from a point of view focusing almost solely on the written records of the Viking attacks in southern part of Great Britain (England). The year 1066 is thus not a definite end-point of the Viking activity, even as for the British Isles, as testified with the repeated delegation of Danish fleets around 1070 (up to 1086 (invasion rumor), almost to the end of the reign of William the Conqueror (d. 1087).

In a sense, the Vikings (the Scandinavians who went out of Scandinavia into naval expedition) turned into the crusaders. Some Danes and Norwegians seemed to join in the First Crusades in the end of the 11th century, and King Sigurd (I) of Norway was in fact the first monarch in Latin West who actually got to the Holy Land (ca. 1110) after the first call of the crusade by Pope Urban II in 1095.

The Eastern Baltic and its people also became the target of the early participants of the Baltic/ Northern Crusades, in which the Danes played an important role.