At this time, it does not appear that a second season of Amazon Panic is in the works. Unfortunately Amazon has not released any official announcement regarding the series, nor any plans for future seasons.
That being said, the series has just begun being streamed to a global audience and it's possible that, depending on its popularity and performance, Amazon could choose to produce another season.
Given that the series is written by Yosuke Kaneko, who has worked on the hugely-successful Running Man manga series, it's possible that Amazon may want to look into producing a second season in the near future. Additionally, Amazon is reportedly already participating in the production of anime works, perhaps indicating that they may be willing to invest in new seasons of Amazon Panic.
If you'd like to follow the progress of Amazon Panic, you may wish to keep an eye on its official Twitter page as well as its official website (https://www.amazonpanic-anime.com/en/). You can also follow Yosuke Kaneko on his Twitter page at @ECCYosukeKaneko.
