The first use of the word "unobtainium" that I find is in the 1930 work "Dechema-Monographien - Volume 39 - Page 12" Reference
...Gewicht gleich Null. Ein Amerikaner hat diesen Werkstoff scherzhafterweise einmal das Element "Unobtainium" genannt.
...Weight equal to zero. An American has jokingly called this material once the element "Unobtainium".
The short story "The Skylark of Space (Amazing Stories 1928) uses a "miracle substance" that is later referred to as "unobtainium" reference but this only uses the concept, I am extremely doubtful that this is the first use of the concept.
The Wikipedia article on Unobtainium (as of this writing) has history going back to the 1950s in use by aerospace engineers. Presumably this use was proceeded by a Science Fiction usage, that brought the term to popularity.
I have found some references suggesting that the 1945 work "Animal Farm" was the first use in English of the word 'Unobtainium", but I have been unable to confirm this.