I think you are:
This element sits on the 47th place in the table of elements, which is a prime number. Silver is clearly our "friend", because many people carry ornamental accesories made of it (jewellery). "Summoned from the vault for destruction and in purity, purify" maybe means when Silver was used as a material to produce bullets, and Silver ions is used nowadays to purify water. When you say "should I join with with another, don't let rain on your parade" maybe you could be talking about the Copper-Silver ionization, used as a water disinfection agent. Finally, "smile big for the camera" might be giving us a clue of the uses of Silver in photography: films, photo printing papers and batteries.
Hope it's not too far!
I am your new friend, ornately adorned,
There's a song which goes "Make new friends, but keep the old. / One is silver, the other is gold." "Ornately adorned" refers to fancy jewelry and accessories.
With a prime seat at the table and manners to match.
Number 47 (a prime) in the periodic table of elements. Fancy place settings have lots of silverware, and knowing which fork to use is important etiquette.
I was summoned from the vault for destruction,
History trivia, silver was borrowed from the U.S. Treasury for use in WWII weaponry, including the Manhattan Project.
And in purity, purify.
Silver nanoparticles are used to purify water.
But should I join with another, don't let it rain on your parade -
These lines are both about silver alloys. Silver iodide is used to seed clouds to produce or attempt to redirect rainfall.
Smile big for the camera!
Silver mercury alloy is used to fill cavities, and film uses silver nitrate or silver halides.
And the title: "A Rhymeless Riddle"
"Silver" does not have any perfect rhymes in English