In Chapter 8 of Lord of the Flies, Jack tries to replace Ralph as chief but no one votes for him. What is at stake for Jack?

This is very clearly a vitally important moment in the
ongoing fued between Jack and Ralph. Jack has shown his own abilities in terms of
successfully hunting pigs and managing the hunting group of the boys, but now the
sightings of the "beast" have given him the excuse he needs to challenge Ralph's
leadership publicly and try to gain the position of leader himself. However, having
publicly challenged Ralph in such a way, there is clearly no going back, and Jack
realises this as the shame and embarrassment of receiving no votes for his proposition
hits him. Note his reaction to the deafening silence that follows his
proposition:

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The silence continued, breathless and heavy and
full of shame. Slowly the red drained from Jack's cheeks, then came back with a painful
rush. He licked his lips and turned his head at an angle, so that his gaze avoided the
embarrassment of linking with another's
eye.

His final response of
"I'm not going to play any longer," symbolises a irrevocable split between the boys,
that Ralph knows can only harm them as they try to co-exist and struggle for survival on
this island. Jack recognises that he does not have the support of the boys, and this is
a blow from which his pride cannot recover.