Is suddenly a linking verb?

No, the word 'suddenly' is not a verb; suddenly is an adverb, a word that modifies a verb (suddenly thought, suddenly appeared).

A linking verb acts somewhat like an equals sign, the object is a form of the subject (Mary is my sister. Mary=sister); or the subject becomes the object (Mary felt cold. Mary->cold). The adverb 'suddenly' can modify the linking verb: Mary suddenly felt cold.