Yes, though when searching for the word "braggardly", you're almost always likely to be presented with the term "braggartly", - the act of being excessively boastful/proud.
Both words mean the same thing. The term braggart, which is synonymous to a boaster, is borrowed from Medieval French bragard "excessively proud," from braguer "to be proud," derivative of brague "liveliness, boastfulness," perhaps originally "to show off clothes, especially breeches," from brague "breeches".
According to Etymonline.com, a website dedicated to providing information on the origin and historical development of words, braggart has been at least influenced by brag, even if, as some claim, it is unrelated to it.
Bragger (arrogant or boastful person) is attested in English from late 14th century and has become practically a variant of this word.
Braggardly also appeared in numerous literary works. A good way to look it up is through Google Books.
Here, you'll find various textual works that make use of this word. One of them is a book called New Catholic World (Volume 47), published in 1888. See link here.
Tip: Some important words from the authors of America's leading and most-trusted provider of language information, Merriam-Webster:
Most general English dictionaries are designed to include only those words that meet certain criteria of usage across wide areas and over extended periods of time (for more details about how words are chosen for dictionary entry, read "How does a word get into a Merriam-Webster dictionary?"). As a result, they may omit words that are still in the process of becoming established, those that are too highly specialized, or those that are so informal that they are rarely documented in professionally edited writing.
The words left out are as real as those that gain entry; the former simply haven't met the criteria for dictionary entry – at least not yet (newer ones may ultimately gain admission to the dictionary's pages if they gain sufficient use).
However, in preparing your own writings, it is worth remembering that the dictionary encompasses the most widely used terms in English. Words that are left out may have usage limited to specific, isolated, or informal contexts, so they should be used carefully.